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PROFESSOR: Last time we
talked about the spin operator

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00:00:26,540 --> 00:00:30,810
pointing in some
particular direction.

10
00:00:30,810 --> 00:00:32,740
There were questions.

11
00:00:32,740 --> 00:00:35,160
In fact, there was
a useful question

12
00:00:35,160 --> 00:00:42,110
that I think I want to begin
the lecture by going back to it.

13
00:00:42,110 --> 00:00:47,740
And this, you received
an email from me.

14
00:00:47,740 --> 00:00:53,880
The notes have an extra section
added to it that is stuff

15
00:00:53,880 --> 00:00:56,140
that I didn't do
in class last time,

16
00:00:56,140 --> 00:01:02,570
but I was told in fact some
of the recitation instructors

17
00:01:02,570 --> 00:01:06,100
did discuss this
matter And I'm going

18
00:01:06,100 --> 00:01:07,920
to say a few words about it.

19
00:01:07,920 --> 00:01:11,990
Now, I do expect you
to read the notes.

20
00:01:11,990 --> 00:01:16,780
So things that you will need for
the homework, all the material

21
00:01:16,780 --> 00:01:20,500
that is in the notes is
material that I kind of

22
00:01:20,500 --> 00:01:23,690
assume you're familiar with.

23
00:01:23,690 --> 00:01:27,118
And you've read it
and understood it.

24
00:01:27,118 --> 00:01:31,560
And I probably don't cover
all what is in the notes,

25
00:01:31,560 --> 00:01:34,440
especially examples
or some things

26
00:01:34,440 --> 00:01:36,000
don't go into so much detail.

27
00:01:36,000 --> 00:01:39,540
But the notes should really be
helping you understand things

28
00:01:39,540 --> 00:01:40,670
well.

29
00:01:40,670 --> 00:01:47,490
So the remark I want to make
is that-- there was a question

30
00:01:47,490 --> 00:01:53,330
last time that better that
we think about it more

31
00:01:53,330 --> 00:01:57,760
deliberately in which we saw
there that Pauli matrices,

32
00:01:57,760 --> 00:02:03,170
sigma 1 squared was equal
to sigma 2 squared equal

33
00:02:03,170 --> 00:02:07,360
to 2 sigma 3 squared
was equal to 1.

34
00:02:07,360 --> 00:02:09,810
Well, that, indeed,
tells you something

35
00:02:09,810 --> 00:02:17,170
important about the
eigenvalues of this matrices.

36
00:02:17,170 --> 00:02:20,280
And it's a general fact.

37
00:02:20,280 --> 00:02:26,585
If you have some matrix M
that satisfies an equation.

38
00:02:26,585 --> 00:02:28,800
Now, let me write an equation.

39
00:02:28,800 --> 00:02:35,560
The matrix M squared plus alpha
M plus beta times the identity

40
00:02:35,560 --> 00:02:37,570
is equal to 0.

41
00:02:37,570 --> 00:02:39,230
This is a matrix equation.

42
00:02:39,230 --> 00:02:43,210
It takes the whole matrix,
square it, add alpha times

43
00:02:43,210 --> 00:02:46,390
the matrix, and then beta
times the identity matrix

44
00:02:46,390 --> 00:02:47,470
is equal to 0.

45
00:02:47,470 --> 00:02:52,440
Suppose you discover that
such an equation holds

46
00:02:52,440 --> 00:02:57,050
for that matrix M. Then,
suppose you are also

47
00:02:57,050 --> 00:03:01,010
asked to find eigenvalues
of this matrix M. So suppose

48
00:03:01,010 --> 00:03:04,200
there is a vector--
that is, an eigenvector

49
00:03:04,200 --> 00:03:06,850
with eigenvalue lambda.

50
00:03:06,850 --> 00:03:09,130
That's what having
an eigenvector

51
00:03:09,130 --> 00:03:12,880
with eigenvalue lambda means.

52
00:03:12,880 --> 00:03:17,300
And you're supposed to calculate
these values of lambda.

53
00:03:17,300 --> 00:03:21,280
So what you do here
is let this equation,

54
00:03:21,280 --> 00:03:25,240
this matrix on the left,
act on the vector v.

55
00:03:25,240 --> 00:03:32,970
So you have M squared plus
alpha M plus beta 1 act

56
00:03:32,970 --> 00:03:41,630
on v. Since the matrix
is 0, it should be 0.

57
00:03:41,630 --> 00:03:44,410
And now you come and
say, well, let's see.

58
00:03:44,410 --> 00:03:48,920
Beta times 1 on v. Well,
that's just beta times

59
00:03:48,920 --> 00:03:53,140
v, the vector v.

60
00:03:53,140 --> 00:03:58,020
Alpha M on v, but M on
v is lambda v. So this

61
00:03:58,020 --> 00:04:06,400
is alpha lambda v. And M squared
on v, as you can imagine,

62
00:04:06,400 --> 00:04:09,100
you act with another M here.

63
00:04:09,100 --> 00:04:10,500
Then you go to this side.

64
00:04:10,500 --> 00:04:14,130
You get lambda Mv, which
is, again, another lambda

65
00:04:14,130 --> 00:04:19,300
times v. So M squared
on v is lambda squared

66
00:04:19,300 --> 00:04:25,960
v. If acts two times on v.

67
00:04:25,960 --> 00:04:28,540
Therefore, this is 0.

68
00:04:28,540 --> 00:04:31,190
And here you have, for
example, that lambda

69
00:04:31,190 --> 00:04:40,020
squared plus alpha lambda
plus beta on v is equal to 0.

70
00:04:40,020 --> 00:04:43,460
Well, v cannot be 0.

71
00:04:43,460 --> 00:04:49,500
Any eigenvector-- by definition,
eigenvectors are not 0 vectors.

72
00:04:49,500 --> 00:04:54,179
You can have 0 eigenvalues
but not 0 eigenvectors.

73
00:04:54,179 --> 00:04:54,970
That doesn't exist.

74
00:04:54,970 --> 00:04:58,720
An eigenvector that
is 0 is a crazy thing

75
00:04:58,720 --> 00:05:01,240
because this would
be 0, and then it

76
00:05:01,240 --> 00:05:05,780
would be-- the eigenvalue
would not be determined.

77
00:05:05,780 --> 00:05:07,190
It just makes no sense.

78
00:05:07,190 --> 00:05:09,190
So v is different from 0.

79
00:05:09,190 --> 00:05:16,300
So you see that lambda squared
plus alpha lambda plus beta is

80
00:05:16,300 --> 00:05:17,820
equal to 0.

81
00:05:17,820 --> 00:05:22,490
And the eigenvalues, any
eigenvalue of this matrix,

82
00:05:22,490 --> 00:05:26,660
must satisfy this equation.

83
00:05:26,660 --> 00:05:29,410
So the eigenvalues
of sigma 1, you

84
00:05:29,410 --> 00:05:33,270
have sigma 1 squared, for
example, is equal to 1.

85
00:05:33,270 --> 00:05:38,880
So the eigenvalues,
any lambda squared

86
00:05:38,880 --> 00:05:42,170
must be equal to
1, the number 1.

87
00:05:44,790 --> 00:05:47,810
And therefore, the
eigenvalues of sigma 1

88
00:05:47,810 --> 00:05:51,260
are possibly plus or minus 1.

89
00:05:51,260 --> 00:05:54,080
We don't know yet.

90
00:05:54,080 --> 00:06:00,730
Could be two 1's, 2 minus
1's, one 1 and one minus 1.

91
00:06:00,730 --> 00:06:07,420
But there's another nice
thing, the trace of sigma 1.

92
00:06:07,420 --> 00:06:09,970
We'll study more the
trace, don't worry.

93
00:06:09,970 --> 00:06:12,430
If you are not that
familiar with it,

94
00:06:12,430 --> 00:06:15,620
it will become
more familiar soon.

95
00:06:15,620 --> 00:06:18,290
The trace of sigma
1 or any matrix

96
00:06:18,290 --> 00:06:20,540
is the sum of elements
in the diagonal.

97
00:06:20,540 --> 00:06:25,290
Sigma 1, if you remember,
was of this form.

98
00:06:25,290 --> 00:06:27,580
Therefore, the trace is 0.

99
00:06:27,580 --> 00:06:33,910
And in fact, the traces of any
of the Pauli matrices are 0.

100
00:06:33,910 --> 00:06:37,010
Another little theorem
of linear algebra

101
00:06:37,010 --> 00:06:43,430
shows that the
trace of a matrix is

102
00:06:43,430 --> 00:06:45,540
equal to the sum of eigenvalues.

103
00:06:45,540 --> 00:06:49,020
So whatever two
eigenvlaues sigma 1 has,

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00:06:49,020 --> 00:06:50,970
they must add up to 0.

105
00:06:50,970 --> 00:06:55,810
Because the trace is 0
and it's equal to the sum

106
00:06:55,810 --> 00:06:57,015
of eigenvalues.

107
00:07:01,310 --> 00:07:04,930
And therefore, if the
eigenvalues can only

108
00:07:04,930 --> 00:07:10,510
be plus or minus 1,
you have the result

109
00:07:10,510 --> 00:07:13,920
that one eigenvalue
must be plus 1.

110
00:07:13,920 --> 00:07:16,290
The other eigenvalue
must be minus 1,

111
00:07:16,290 --> 00:07:19,480
is the only way you
can get that to work.

112
00:07:19,480 --> 00:07:33,451
So two sigma 1 eigenvalues of
sigma 1 are plus 1 and minus 1.

113
00:07:33,451 --> 00:07:34,700
Those are the two eigenvalues.

114
00:07:38,390 --> 00:07:43,980
So in that section
as well, there's

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00:07:43,980 --> 00:07:48,260
some discussion about properties
of the Pauli matrices.

116
00:07:48,260 --> 00:07:54,390
And two basic properties
of Pauli matrices

117
00:07:54,390 --> 00:07:56,680
are the following.

118
00:07:56,680 --> 00:08:00,780
Remember that the spin
matrices, the spin operators,

119
00:08:00,780 --> 00:08:05,220
are h bar over 2 times
the Pauli matrices.

120
00:08:05,220 --> 00:08:09,850
And the spin operators had the
algebra for angular momentum.

121
00:08:09,850 --> 00:08:12,510
So from the algebra
of angular momentum

122
00:08:12,510 --> 00:08:23,820
that says that Si Sj is equal
to i h bar epsilon i j k Sk,

123
00:08:23,820 --> 00:08:29,860
you deduce after plugging
this that sigma i sigma

124
00:08:29,860 --> 00:08:36,320
j is 2i epsilon i j k sigma k.

125
00:08:45,130 --> 00:08:47,480
Moreover, there's
another nice property

126
00:08:47,480 --> 00:08:52,800
of the Pauli matrices having
to deal with anticommutators.

127
00:08:52,800 --> 00:08:58,150
If you do experimentally
try multiplying

128
00:08:58,150 --> 00:09:01,710
Pauli matrices,
sigma 1 and sigma 2,

129
00:09:01,710 --> 00:09:04,970
you will find out that if you
compare it with sigma 2 sigma

130
00:09:04,970 --> 00:09:06,750
1, it's different.

131
00:09:06,750 --> 00:09:09,950
Of course, it's not the same.

132
00:09:09,950 --> 00:09:11,460
These matrices don't commute.

133
00:09:11,460 --> 00:09:15,000
But they actually-- while
they fail to commute,

134
00:09:15,000 --> 00:09:17,470
they still fail to
commute in a nice way.

135
00:09:17,470 --> 00:09:21,530
Actually, these are
minus each other.

136
00:09:21,530 --> 00:09:28,130
So in fact, sigma 1 sigma 2 plus
sigma 2 sigma 1 is equal to 0.

137
00:09:28,130 --> 00:09:31,470
And by this, we mean
that they anticommute.

138
00:09:31,470 --> 00:09:35,050
And we have a brief
way of calling this.

139
00:09:35,050 --> 00:09:38,580
When this sign was a minus,
it was called the commutator.

140
00:09:38,580 --> 00:09:42,460
When this is a plus, it's
called an anticommutator.

141
00:09:42,460 --> 00:09:48,930
So the anticommutator of sigma
1 with sigma 2 is equal to 0.

142
00:09:48,930 --> 00:09:53,730
Anticommutator defined
in general by A,

143
00:09:53,730 --> 00:09:58,090
B. Two operators is AB plus BA.

144
00:10:01,100 --> 00:10:03,540
And as you will
read in the notes,

145
00:10:03,540 --> 00:10:06,955
a little more analysis
shows that, in fact,

146
00:10:06,955 --> 00:10:10,290
the anticommutator of
sigma i and sigma j

147
00:10:10,290 --> 00:10:16,300
has a nice formula,
which is 2 delta ij times

148
00:10:16,300 --> 00:10:19,490
the unit matrix, the
2 by 2 unit matrix.

149
00:10:25,640 --> 00:10:29,860
With this result, you
get a general formula.

150
00:10:29,860 --> 00:10:36,190
Any product of two operators,
AB, you can write as 1/2

151
00:10:36,190 --> 00:10:41,569
of the anticommutator plus
1-- no, 1/2 of the commutator

152
00:10:41,569 --> 00:10:42,860
plus 1/2 of the anticommutator.

153
00:10:46,510 --> 00:10:50,450
Expand it out, that
right-hand side,

154
00:10:50,450 --> 00:10:52,520
and you will see
quite quickly this

155
00:10:52,520 --> 00:10:55,980
is true for any two operators.

156
00:10:55,980 --> 00:11:00,450
This has AB minus BA
and this has AB plus BA.

157
00:11:00,450 --> 00:11:05,370
The BA term cancels and the
AB terms are [INAUDIBLE].

158
00:11:05,370 --> 00:11:13,140
So sigma i sigma j
would be equal to 1/2.

159
00:11:13,140 --> 00:11:15,450
And then they put down
the anticommutator first.

160
00:11:15,450 --> 00:11:19,700
So you get delta ij
times the identity, which

161
00:11:19,700 --> 00:11:22,960
is 1/2 of the
anticommutator plus 1/2

162
00:11:22,960 --> 00:11:30,129
of the commutator, which
is i epsilon i j k sigma k.

163
00:11:35,510 --> 00:11:37,530
It's a very useful formula.

164
00:11:40,930 --> 00:11:46,130
In order to make those
formulas look neater,

165
00:11:46,130 --> 00:11:54,400
we invent a notation in which
we think of sigma as a triplet--

166
00:11:54,400 --> 00:11:58,440
sigma 1, sigma 2, and sigma 3.

167
00:11:58,440 --> 00:12:04,470
And then we have vectors,
like a-- normal vectors,

168
00:12:04,470 --> 00:12:06,170
components a1, a2, a3.

169
00:12:09,080 --> 00:12:17,520
And then we have a dot
sigma must be defined.

170
00:12:17,520 --> 00:12:19,480
Well, there's an
obvious definition

171
00:12:19,480 --> 00:12:22,180
of what this should
mean, but it's not

172
00:12:22,180 --> 00:12:24,390
something you're accustomed to.

173
00:12:24,390 --> 00:12:27,340
And one should pause
before saying this.

174
00:12:27,340 --> 00:12:31,440
You're having a normal
vector, a triplet of numbers,

175
00:12:31,440 --> 00:12:34,780
multiplied by a
triplet of matrices,

176
00:12:34,780 --> 00:12:38,370
or a triplet of operators.

177
00:12:38,370 --> 00:12:41,930
Since numbers commute
with matrices,

178
00:12:41,930 --> 00:12:45,000
the order in which you
write this doesn't matter.

179
00:12:45,000 --> 00:12:49,135
But this is defined
to be a1 sigma 1

180
00:12:49,135 --> 00:12:52,930
plus a2 sigma 2 plus a3 sigma 3.

181
00:12:56,250 --> 00:13:02,990
This can be written as ai
sigma i with our repeated index

182
00:13:02,990 --> 00:13:06,550
convention that you sum
over the possibilities.

183
00:13:06,550 --> 00:13:10,360
So here is what you're
supposed to do here

184
00:13:10,360 --> 00:13:14,070
to maybe interpret
this equation nicely.

185
00:13:14,070 --> 00:13:18,850
You multiply this
equation n by ai bj.

186
00:13:21,480 --> 00:13:22,930
Now, these are numbers.

187
00:13:22,930 --> 00:13:24,210
These are matrices.

188
00:13:24,210 --> 00:13:27,810
I better not change this
order, but I can certainly,

189
00:13:27,810 --> 00:13:34,590
by multiplying that way, I
have ai sigma i bj sigma j

190
00:13:34,590 --> 00:13:46,070
equals 2 ai bj delta ij
times the matrix 1 plus i

191
00:13:46,070 --> 00:13:52,740
epsilon i j k ai bj sigma k.

192
00:14:01,660 --> 00:14:02,860
Now, what?

193
00:14:02,860 --> 00:14:06,490
Well, write it in terms
of things that look neat.

194
00:14:06,490 --> 00:14:11,420
a dot sigma, that's a matrix.

195
00:14:11,420 --> 00:14:15,010
This whole thing is a matrix
multiplied by the matrix

196
00:14:15,010 --> 00:14:20,750
b dot sigma gives you--

197
00:14:20,750 --> 00:14:29,580
Well, ai bj delta ij, this
delta ij forces j to become i.

198
00:14:29,580 --> 00:14:34,460
In other words, you can replace
these two terms by just bi.

199
00:14:34,460 --> 00:14:36,980
And then you have ai bi.

200
00:14:36,980 --> 00:14:39,934
So this is twice.

201
00:14:39,934 --> 00:14:42,640
I don't know why I have a 2.

202
00:14:42,640 --> 00:14:44,780
No 2.

203
00:14:44,780 --> 00:14:48,300
There was no 2 there, sorry.

204
00:14:48,300 --> 00:14:49,650
So what do we get here?

205
00:14:49,650 --> 00:14:54,300
We get a dot b, the dot product.

206
00:14:54,300 --> 00:14:57,050
This is a normal dot product.

207
00:14:57,050 --> 00:15:02,690
This is just a number
times 1 plus i.

208
00:15:02,690 --> 00:15:05,580
Now, what is this thing?

209
00:15:05,580 --> 00:15:09,740
You should try to remember
how the epsilon tensor can

210
00:15:09,740 --> 00:15:11,880
be used to do cross products.

211
00:15:11,880 --> 00:15:16,820
This, there's just one
free index, the index k.

212
00:15:16,820 --> 00:15:19,000
So this must be
some sort of vector.

213
00:15:19,000 --> 00:15:24,090
And in fact, if you try the
definition of epsilon and look

214
00:15:24,090 --> 00:15:26,670
in detail what this
is, you will find

215
00:15:26,670 --> 00:15:35,410
that this is nothing but
the k component of a dot b.

216
00:15:35,410 --> 00:15:37,280
The k-- so I'll write it here.

217
00:15:37,280 --> 00:15:42,710
This is a cross b sub k.

218
00:15:42,710 --> 00:15:47,230
But now you have a cross
b sub k times sigma k.

219
00:15:47,230 --> 00:15:52,255
So this is the same as
a cross b dot sigma.

220
00:15:55,960 --> 00:16:03,375
And here you got a pretty nice
equation for Pauli matrices.

221
00:16:06,490 --> 00:16:10,930
It expresses the general
product of Pauli matrices

222
00:16:10,930 --> 00:16:14,090
in somewhat geometric terms.

223
00:16:14,090 --> 00:16:25,490
So if you take, for
example here, an operator.

224
00:16:25,490 --> 00:16:26,420
No.

225
00:16:26,420 --> 00:16:29,640
If you take, for
example, a equals

226
00:16:29,640 --> 00:16:38,680
b equal to a unit vector,
then what do we get?

227
00:16:38,680 --> 00:16:41,717
You get n dot sigma squared.

228
00:16:46,250 --> 00:16:48,530
And here you have
the dot product of n

229
00:16:48,530 --> 00:16:50,210
with n, which is 1.

230
00:16:50,210 --> 00:16:53,420
So this is 1.

231
00:16:53,420 --> 00:16:56,960
And the cross product of two
equal vectors, of course,

232
00:16:56,960 --> 00:17:03,970
is 0 so you get
this, which is nice.

233
00:17:03,970 --> 00:17:05,380
Why is this useful?

234
00:17:05,380 --> 00:17:11,760
It's because with this identity,
you can understand better

235
00:17:11,760 --> 00:17:15,410
the operator S hat
n that we introduced

236
00:17:15,410 --> 00:17:22,950
last time, which was n
dot the spin triplet.

237
00:17:22,950 --> 00:17:28,990
So nx, sx, ny, sy, nz, sc.

238
00:17:28,990 --> 00:17:30,370
So what is this?

239
00:17:30,370 --> 00:17:35,260
This is h bar over
2 and dot sigma.

240
00:17:40,320 --> 00:17:42,310
And let's square this.

241
00:17:42,310 --> 00:17:46,510
So Sn vector squared.

242
00:17:46,510 --> 00:17:52,420
This matrix squared would be
h bar over 2 squared times

243
00:17:52,420 --> 00:17:55,085
n dot sigma squared, which is 1.

244
00:18:04,600 --> 00:18:06,330
And sigma squared is 1.

245
00:18:06,330 --> 00:18:13,540
Therefore, this spin operator
along the n direction squares

246
00:18:13,540 --> 00:18:16,620
to h bar r squared
over 2 times 1.

247
00:18:16,620 --> 00:18:25,500
Now, the trace of this
Sn operator is also 0.

248
00:18:25,500 --> 00:18:26,860
Why?

249
00:18:26,860 --> 00:18:29,390
Because the trace
means that you're

250
00:18:29,390 --> 00:18:32,280
going to sum the
elements in the diagonal.

251
00:18:32,280 --> 00:18:36,260
Well, you have a sum
of matrices here.

252
00:18:36,260 --> 00:18:40,310
And therefore, you will have
to sum the diagonals of each.

253
00:18:40,310 --> 00:18:43,690
But each of the
sigmas has 0 trace.

254
00:18:43,690 --> 00:18:46,380
We wrote it there.

255
00:18:46,380 --> 00:18:48,310
Trace of sigma 1 is 0.

256
00:18:48,310 --> 00:18:53,660
All the Pauli matrices have
0 trace, so this has 0 trace.

257
00:18:53,660 --> 00:18:57,930
So you have these two relations.

258
00:18:57,930 --> 00:19:03,250
And again, this tells you that
the eigenvalues of this matrix

259
00:19:03,250 --> 00:19:07,210
can be plus minus h bar over 2.

260
00:19:07,210 --> 00:19:10,010
Because the eigenvalues
satisfy the same equation

261
00:19:10,010 --> 00:19:11,250
as the matrix.

262
00:19:11,250 --> 00:19:14,460
Therefor,e plus
minus h bar over 2.

263
00:19:14,460 --> 00:19:19,090
And this one says that the
eigenvalues add up to 0.

264
00:19:19,090 --> 00:19:29,070
So the eigenvalues of S hat n
vector are plus h bar over 2

265
00:19:29,070 --> 00:19:31,410
and minus h bar over 2.

266
00:19:31,410 --> 00:19:36,170
We did that last time, but we do
that by just taking that matrix

267
00:19:36,170 --> 00:19:37,820
and finding the eigenvalues.

268
00:19:37,820 --> 00:19:44,340
But this shows that its
property is almost manifest.

269
00:19:44,340 --> 00:19:47,180
And this is fundamental
for the interpretation

270
00:19:47,180 --> 00:19:49,310
of this operator.

271
00:19:49,310 --> 00:19:50,310
Why?

272
00:19:50,310 --> 00:19:54,110
Well, we saw that if n
points along the z-direction,

273
00:19:54,110 --> 00:19:56,340
it becomes the operator sz.

274
00:19:56,340 --> 00:19:58,490
If it points about
the x-direction,

275
00:19:58,490 --> 00:20:00,810
it becomes the operator sx.

276
00:20:00,810 --> 00:20:03,890
If it points along
y, it becomes sy.

277
00:20:03,890 --> 00:20:07,460
But in an arbitrary
direction, it's a funny thing.

278
00:20:07,460 --> 00:20:10,410
But it still has
the key property.

279
00:20:10,410 --> 00:20:14,200
If you measured the spin
along an arbitrary direction,

280
00:20:14,200 --> 00:20:19,740
you should find only plus h bar
over 2 or minus h bar over 2.

281
00:20:19,740 --> 00:20:23,320
Because after all, the
universe is isotopic.

282
00:20:23,320 --> 00:20:25,560
It doesn't depend on direction.

283
00:20:25,560 --> 00:20:27,500
So a spin one-half particle.

284
00:20:27,500 --> 00:20:30,660
If you find out that whenever
you measure the z component,

285
00:20:30,660 --> 00:20:33,250
it's either plus
minus h bar over 2.

286
00:20:33,250 --> 00:20:35,355
Well, when you
measure any direction,

287
00:20:35,355 --> 00:20:39,120
it should be plus
minus h bar over 2.

288
00:20:39,120 --> 00:20:42,940
And this shows that this
operator has those eigenvalues.

289
00:20:42,940 --> 00:20:47,570
And therefore, it makes sense
that this is the operator

290
00:20:47,570 --> 00:20:52,180
that measures spins in
an arbitrary direction.

291
00:20:52,180 --> 00:20:56,595
There's a little more
of an aside in there,

292
00:20:56,595 --> 00:20:58,610
in the notes about
something that

293
00:20:58,610 --> 00:21:02,590
will be useful and fun to do.

294
00:21:02,590 --> 00:21:04,830
And it corresponds to
the case in which you

295
00:21:04,830 --> 00:21:10,150
have two triplets of
operators-- x1, x2, x3.

296
00:21:10,150 --> 00:21:12,700
These are operators now.

297
00:21:12,700 --> 00:21:18,945
And y equal y1, y2, y3.

298
00:21:22,120 --> 00:21:24,160
Two triplets of operators.

299
00:21:24,160 --> 00:21:29,585
So you define the dot
product of these two triplets

300
00:21:29,585 --> 00:21:36,149
as xi yi summed.

301
00:21:36,149 --> 00:21:37,065
That's the definition.

302
00:21:41,920 --> 00:21:45,190
Now, the dot product of
two triplets of operators

303
00:21:45,190 --> 00:21:48,680
defined that way
may not commute.

304
00:21:48,680 --> 00:21:52,190
Because the operators x
and y may not commute.

305
00:21:52,190 --> 00:21:58,150
So this new dot product
of both phase operators

306
00:21:58,150 --> 00:22:01,380
is not commutative-- probably.

307
00:22:01,380 --> 00:22:03,800
It may happen that
these operators commute,

308
00:22:03,800 --> 00:22:07,630
in which case x dot y
is equal to y dot x.

309
00:22:07,630 --> 00:22:10,620
Similarly, you can
define the cross product

310
00:22:10,620 --> 00:22:14,000
of these two things.

311
00:22:14,000 --> 00:22:23,610
And the k-th component is
epsilon i j k xi yj like this.

312
00:22:26,830 --> 00:22:30,565
Just like you would define
it for two number vectors.

313
00:22:33,460 --> 00:22:36,860
Now, what do you know about
the cross product in general?

314
00:22:36,860 --> 00:22:37,875
It's anti-symmetric.

315
00:22:37,875 --> 00:22:40,400
A cross B is equal
to minus B cross A.

316
00:22:40,400 --> 00:22:48,290
But this one won't be
because the operators x and y

317
00:22:48,290 --> 00:22:49,490
may not commute.

318
00:22:49,490 --> 00:22:56,490
Even x cross x may be nonzero.

319
00:22:56,490 --> 00:23:00,180
So one thing I will ask you
to compute in the homework

320
00:23:00,180 --> 00:23:03,180
is not a long calculation.

321
00:23:03,180 --> 00:23:04,150
It's three lines.

322
00:23:04,150 --> 00:23:14,300
But what is S cross S equal to?

323
00:23:14,300 --> 00:23:15,040
Question there?

324
00:23:15,040 --> 00:23:15,956
AUDIENCE: [INAUDIBLE].

325
00:23:19,770 --> 00:23:21,860
PROFESSOR: Yes, it's
the sum [INAUDIBLE].

326
00:23:21,860 --> 00:23:24,710
Just in the same
way that here you're

327
00:23:24,710 --> 00:23:29,270
summing over i's and j's to
produce the cross product.

328
00:23:29,270 --> 00:23:31,810
So whenever an
index is repeated,

329
00:23:31,810 --> 00:23:34,780
we'll assume it's summed.

330
00:23:34,780 --> 00:23:37,920
And when it is not summed,
I will put to the right,

331
00:23:37,920 --> 00:23:41,110
not summed explicitly--
the words-.

332
00:23:41,110 --> 00:23:45,040
Because in some
occasions, it matters.

333
00:23:45,040 --> 00:23:46,820
So how much is this?

334
00:23:46,820 --> 00:23:50,840
It will involve i, h
bar, and something.

335
00:23:50,840 --> 00:23:53,790
And you will try to
find out what this is.

336
00:23:53,790 --> 00:23:56,930
It's a cute thing.

337
00:23:56,930 --> 00:23:59,533
All right, any other questions?

338
00:24:07,880 --> 00:24:09,270
More questions?

339
00:24:09,270 --> 00:24:09,770
Nope.

340
00:24:13,120 --> 00:24:13,800
OK.

341
00:24:13,800 --> 00:24:16,950
So now, finally,
we get to that part

342
00:24:16,950 --> 00:24:20,980
of the course that has to
do with linear algebra.

343
00:24:20,980 --> 00:24:25,050
And I'm going to
do an experiment.

344
00:24:25,050 --> 00:24:27,710
I'm going to do it
differently than I did it

345
00:24:27,710 --> 00:24:28,815
in the previous years.

346
00:24:32,450 --> 00:24:35,720
There is this nice book.

347
00:24:35,720 --> 00:24:37,710
It's here.

348
00:24:37,710 --> 00:24:40,090
I don't know if you
can read from that far,

349
00:24:40,090 --> 00:24:47,430
but it has a pretty-- you might
almost say an arrogant title.

350
00:24:47,430 --> 00:24:54,280
It says, Linear Algebra
Done Right by Sheldon Axler.

351
00:24:54,280 --> 00:25:00,140
This is the book, actually,
MIT's course 18.700 of linear

352
00:25:00,140 --> 00:25:02,560
algebra uses.

353
00:25:02,560 --> 00:25:05,450
And when you first get the
book that looks like that,

354
00:25:05,450 --> 00:25:08,340
you read it and open--
I'm going to show you

355
00:25:08,340 --> 00:25:11,760
that this is not that well done.

356
00:25:11,760 --> 00:25:15,510
But actually, I think
it's actually true.

357
00:25:15,510 --> 00:25:18,200
The title is not a lie.

358
00:25:18,200 --> 00:25:21,530
It's really done right.

359
00:25:21,530 --> 00:25:26,370
I actually wish I had learned
linear algebra this way.

360
00:25:26,370 --> 00:25:29,500
It may be a little
difficult if you've never

361
00:25:29,500 --> 00:25:32,550
done any linear algebra.

362
00:25:32,550 --> 00:25:34,780
You don't know what
the matrix is--

363
00:25:34,780 --> 00:25:36,640
I don't think that's
the case anybody here.

364
00:25:36,640 --> 00:25:41,580
A determinant, or eigenvalue.

365
00:25:41,580 --> 00:25:43,670
If you never heard
any of those words,

366
00:25:43,670 --> 00:25:46,050
this might be a little hard.

367
00:25:46,050 --> 00:25:48,110
But if you've heard
those words and you've

368
00:25:48,110 --> 00:25:50,930
had a little linear
algebra, this is quite nice.

369
00:25:50,930 --> 00:25:54,125
Now, this book has
also a small problem.

370
00:25:57,000 --> 00:25:59,930
Unless you study
it seriously, it's

371
00:25:59,930 --> 00:26:03,750
not all that easy to grab
results that you need from it.

372
00:26:03,750 --> 00:26:05,860
You have to study it.

373
00:26:05,860 --> 00:26:08,180
So I don't know if
it might help you

374
00:26:08,180 --> 00:26:10,420
or not during this semester.

375
00:26:10,420 --> 00:26:12,290
It may.

376
00:26:12,290 --> 00:26:14,060
It's not necessary to get it.

377
00:26:14,060 --> 00:26:15,620
Absolutely not.

378
00:26:15,620 --> 00:26:19,090
But it is quite lovely.

379
00:26:19,090 --> 00:26:21,630
And the emphasis is
quite interesting.

380
00:26:21,630 --> 00:26:25,780
It really begins from
very basic things

381
00:26:25,780 --> 00:26:28,520
and logically
develops everything

382
00:26:28,520 --> 00:26:31,520
and asks at every point
the right questions.

383
00:26:31,520 --> 00:26:32,580
It's quite nice.

384
00:26:32,580 --> 00:26:36,860
So what I'm going to do
is-- inspired by that,

385
00:26:36,860 --> 00:26:42,220
I want to introduce some of
the linear algebra little

386
00:26:42,220 --> 00:26:42,960
by little.

387
00:26:42,960 --> 00:26:45,910
And I don't know very
well how this will go.

388
00:26:45,910 --> 00:26:47,630
Maybe there's too much detail.

389
00:26:47,630 --> 00:26:52,120
Maybe it's a lot of
detail, but not enough so

390
00:26:52,120 --> 00:26:53,530
it's not all that great.

391
00:26:53,530 --> 00:26:55,750
I don't know, you
will have to tell me.

392
00:26:58,490 --> 00:27:01,130
But we'll try to get
some ideas clear.

393
00:27:01,130 --> 00:27:03,990
And the reason I want
to get some ideas clear

394
00:27:03,990 --> 00:27:08,880
is that good books
on this subject

395
00:27:08,880 --> 00:27:12,070
allow you to understand
how much structure you

396
00:27:12,070 --> 00:27:16,680
have to put in a vector space
to define certain things.

397
00:27:16,680 --> 00:27:20,470
And unless you do
this carefully,

398
00:27:20,470 --> 00:27:25,730
you probably miss some
of the basic things.

399
00:27:25,730 --> 00:27:30,240
Like many physicists
don't quite realize

400
00:27:30,240 --> 00:27:32,980
that talking about the
matrix representation,

401
00:27:32,980 --> 00:27:34,970
you don't need brass
and [INAUDIBLE]

402
00:27:34,970 --> 00:27:38,040
to talk about the matrix
representation of an operator.

403
00:27:38,040 --> 00:27:40,670
At first sight, it seems
like you'd need it,

404
00:27:40,670 --> 00:27:42,970
but you actually don't.

405
00:27:42,970 --> 00:27:46,640
Then, the differences between
a complex and a vector space--

406
00:27:46,640 --> 00:27:51,300
complex and a real vector
space become much clearer

407
00:27:51,300 --> 00:27:53,570
if you take your time
to understand it.

408
00:27:53,570 --> 00:27:55,320
They are very different.

409
00:27:55,320 --> 00:27:57,740
And in a sense,
complex vector spaces

410
00:27:57,740 --> 00:28:01,350
are more powerful, more
elegant, have stronger results.

411
00:28:04,260 --> 00:28:07,870
So anyway, it's enough
of an introduction.

412
00:28:07,870 --> 00:28:10,710
Let's see how we do.

413
00:28:10,710 --> 00:28:14,870
And let's just begin
there for our story.

414
00:28:14,870 --> 00:28:21,770
So we begin with vector
spaces and dimensionality.

415
00:28:21,770 --> 00:28:22,510
Yes.

416
00:28:22,510 --> 00:28:24,438
AUDIENCE: Quick question.

417
00:28:24,438 --> 00:28:28,776
The length between
the trace of matrix

418
00:28:28,776 --> 00:28:33,450
equals 0 and [INAUDIBLE] is
proportional to the identity.

419
00:28:33,450 --> 00:28:35,997
One is the product of
the eigenvalues is 1

420
00:28:35,997 --> 00:28:39,893
and the other one was
the sum is equal to 0.

421
00:28:39,893 --> 00:28:42,328
Are those two statements
related causally,

422
00:28:42,328 --> 00:28:44,710
or are they just separate
statements [INAUDIBLE]?

423
00:28:44,710 --> 00:28:46,210
PROFESSOR: OK, the
question is, what

424
00:28:46,210 --> 00:28:48,560
is the relation between
these two statements?

425
00:28:48,560 --> 00:28:50,570
Those are separate observations.

426
00:28:50,570 --> 00:28:53,050
One does not imply the other.

427
00:28:53,050 --> 00:28:56,490
You can have matrices that
square to the identity,

428
00:28:56,490 --> 00:29:00,290
like the identity itself,
and don't have 0 trace.

429
00:29:00,290 --> 00:29:02,590
So these are
separate properties.

430
00:29:02,590 --> 00:29:06,785
This tells us that
the eigenvalue squared

431
00:29:06,785 --> 00:29:10,010
are h bar over 2.

432
00:29:10,010 --> 00:29:14,280
And this one tells me that
lambda 1 plus lambda 2--

433
00:29:14,280 --> 00:29:16,780
there are two
eigenvalues-- are 0.

434
00:29:16,780 --> 00:29:20,250
So from here, you deduce
that the eigenvalues

435
00:29:20,250 --> 00:29:22,350
could be plus
minus h bar over 2.

436
00:29:22,350 --> 00:29:25,450
And in fact, have to be
plus minus h bar over 2.

437
00:29:28,520 --> 00:29:32,850
All right, so let's
talk about vector spaces

438
00:29:32,850 --> 00:29:35,590
and dimensionality.

439
00:29:35,590 --> 00:29:39,273
Spaces and dimensionality.

440
00:29:47,260 --> 00:29:50,480
So why do we care about this?

441
00:29:50,480 --> 00:29:53,060
Because the end result
of our discussion

442
00:29:53,060 --> 00:29:56,600
is that the states
of a physical system

443
00:29:56,600 --> 00:30:01,050
are vectors in a
complex vector space.

444
00:30:01,050 --> 00:30:04,820
That's, in a sense, the
result we're going to get.

445
00:30:04,820 --> 00:30:09,520
Observables, moreover,
are linear operators

446
00:30:09,520 --> 00:30:11,420
on those vector spaces.

447
00:30:11,420 --> 00:30:15,070
So we need to understand what
are complex vector spaces, what

448
00:30:15,070 --> 00:30:18,930
linear operators on them mean.

449
00:30:18,930 --> 00:30:22,650
So as I said,
complex vector spaces

450
00:30:22,650 --> 00:30:26,350
have subtle properties that make
them different from real vector

451
00:30:26,350 --> 00:30:28,840
spaces and we want
to appreciate that.

452
00:30:28,840 --> 00:30:32,110
In a vector space,
what do you have?

453
00:30:32,110 --> 00:30:36,830
You have vectors and
you have numbers.

454
00:30:36,830 --> 00:30:39,280
So the two things must exist.

455
00:30:39,280 --> 00:30:43,710
The numbers could be the
real numbers, in which case

456
00:30:43,710 --> 00:30:46,310
we're talking about
the real vector space.

457
00:30:46,310 --> 00:30:49,680
And the numbers could be
complex numbers, in which case

458
00:30:49,680 --> 00:30:52,530
we're talking about the
complex vector space.

459
00:30:52,530 --> 00:30:58,670
We don't say the vectors are
real, or complex, or imaginary.

460
00:30:58,670 --> 00:31:03,620
We just say there are vectors
and there are numbers.

461
00:31:03,620 --> 00:31:08,930
Now, the vectors can be
added and the numbers

462
00:31:08,930 --> 00:31:12,180
can be multiplied by
vectors to give vectors.

463
00:31:12,180 --> 00:31:15,500
That's basically
what is happening.

464
00:31:15,500 --> 00:31:20,320
Now, these numbers can
be real or complex.

465
00:31:20,320 --> 00:31:26,933
And the numbers-- so there
are vectors and numbers.

466
00:31:30,050 --> 00:31:33,550
And we will focus on
just either real numbers

467
00:31:33,550 --> 00:31:36,720
or complex numbers,
but either one.

468
00:31:36,720 --> 00:31:41,400
So these sets of
numbers form what

469
00:31:41,400 --> 00:31:43,770
is called in
mathematics a field.

470
00:31:43,770 --> 00:31:46,970
So I will not define the field.

471
00:31:46,970 --> 00:31:52,430
But a field-- use the
letter F for field.

472
00:31:52,430 --> 00:31:53,720
And our results.

473
00:31:53,720 --> 00:31:58,220
I will state results whenever--
it doesn't matter whether it's

474
00:31:58,220 --> 00:32:01,460
real or complex, I
may use the letter F

475
00:32:01,460 --> 00:32:05,480
to say the numbers are in F.
And you say real or complex.

476
00:32:10,260 --> 00:32:12,250
What is a vector space?

477
00:32:12,250 --> 00:32:22,310
So the vector space,
V. Vector space, V,

478
00:32:22,310 --> 00:32:36,410
is a set of vectors with an
operation called addition--

479
00:32:36,410 --> 00:32:54,160
and we represent it as plus--
that assigns a vector u plus v

480
00:32:54,160 --> 00:33:05,320
in the vector space when u and
v belong to the vector space.

481
00:33:05,320 --> 00:33:08,170
So for any u and v
in the vector space,

482
00:33:08,170 --> 00:33:14,170
there's a rule called addition
that assigns another vector.

483
00:33:14,170 --> 00:33:18,410
This also means that this
space is closed under addition.

484
00:33:18,410 --> 00:33:21,500
That is, you cannot get out
of the vector space by adding

485
00:33:21,500 --> 00:33:22,640
vectors.

486
00:33:22,640 --> 00:33:25,930
The vector space must
contain a set that

487
00:33:25,930 --> 00:33:28,390
is consistent in that
you can add vectors

488
00:33:28,390 --> 00:33:29,896
and you're always there.

489
00:33:29,896 --> 00:33:31,104
And there's a multiplication.

490
00:33:33,732 --> 00:33:49,420
And a scalar
multiplication by elements

491
00:33:49,420 --> 00:34:02,130
of the numbers of F such
that a, which is a number,

492
00:34:02,130 --> 00:34:07,110
times v belongs to
the vector space

493
00:34:07,110 --> 00:34:16,889
when a belongs to the numbers
and v belongs to the vectors.

494
00:34:16,889 --> 00:34:19,050
So every time you
have a vector, you

495
00:34:19,050 --> 00:34:22,730
can multiply by those
numbers and the result

496
00:34:22,730 --> 00:34:25,940
of that multiplication
is another vector.

497
00:34:25,940 --> 00:34:31,610
So we say the space is also
closed under multiplication.

498
00:34:31,610 --> 00:34:33,800
Now, these properties
exist, but they

499
00:34:33,800 --> 00:34:37,710
must-- these operations
exist, but they

500
00:34:37,710 --> 00:34:39,870
must satisfy the
following properties.

501
00:34:39,870 --> 00:34:41,809
So the definition
is not really over.

502
00:34:45,590 --> 00:34:49,753
These operations satisfy--

503
00:34:54,520 --> 00:34:56,030
1.

504
00:34:56,030 --> 00:34:59,770
u plus v is equal to v plus u.

505
00:34:59,770 --> 00:35:03,150
The order doesn't matter
how you sum vectors.

506
00:35:03,150 --> 00:35:07,830
And here, u and v in V.

507
00:35:07,830 --> 00:35:09,490
2.

508
00:35:09,490 --> 00:35:10,480
Associative.

509
00:35:10,480 --> 00:35:21,130
So u plus v plus w is
equal to u plus v plus w.

510
00:35:21,130 --> 00:35:30,950
Moreover, two numbers a times b
times v is the same as a times

511
00:35:30,950 --> 00:35:33,420
bv.

512
00:35:33,420 --> 00:35:35,624
You can add with the
first number on the vector

513
00:35:35,624 --> 00:35:36,790
and you add with the second.

514
00:35:41,920 --> 00:35:42,420
3.

515
00:35:45,400 --> 00:35:52,245
There is an additive identity.

516
00:35:55,930 --> 00:35:57,130
And that is what?

517
00:35:57,130 --> 00:36:01,450
It's a vector 0 belonging
to the vector space.

518
00:36:01,450 --> 00:36:03,370
I could write an arrow.

519
00:36:03,370 --> 00:36:07,570
But actually, for
some reason they just

520
00:36:07,570 --> 00:36:09,280
don't like to write
it because they say

521
00:36:09,280 --> 00:36:11,780
it's always ambiguous
whether you're

522
00:36:11,780 --> 00:36:15,910
talking about the 0
number or the 0 vector.

523
00:36:15,910 --> 00:36:18,050
We do have that problem
also in the notation

524
00:36:18,050 --> 00:36:19,480
in quantum mechanics.

525
00:36:19,480 --> 00:36:28,950
But here it is, here is
a 0 vector such that 0

526
00:36:28,950 --> 00:36:37,812
plus any vector v is equal to v.

527
00:36:37,812 --> 00:36:40,200
4.

528
00:36:40,200 --> 00:36:42,850
Well, in the field,
in the set of numbers,

529
00:36:42,850 --> 00:36:47,420
there's the number 1, which
multiplied by any other number

530
00:36:47,420 --> 00:36:49,790
keeps that number.

531
00:36:49,790 --> 00:36:58,210
So the number 1 that
belongs to the field

532
00:36:58,210 --> 00:37:08,050
satisfies that 1 times any
vector is equal to the vector.

533
00:37:08,050 --> 00:37:13,540
So we declare that that number
multiplied by other numbers

534
00:37:13,540 --> 00:37:14,780
is an identity.

535
00:37:14,780 --> 00:37:17,302
[INAUDIBLE] identity
also multiplying vectors.

536
00:37:17,302 --> 00:37:18,385
Yes, there was a question.

537
00:37:18,385 --> 00:37:21,120
AUDIENCE: [INAUDIBLE].

538
00:37:21,120 --> 00:37:24,945
PROFESSOR: There is
an additive identity.

539
00:37:24,945 --> 00:37:31,130
Additive identity, the 0 vector.

540
00:37:31,130 --> 00:37:35,532
Finally, distributive laws.

541
00:37:35,532 --> 00:37:37,850
No.

542
00:37:37,850 --> 00:37:38,690
One second.

543
00:37:38,690 --> 00:37:46,290
One, two, three--
the zero vector.

544
00:37:46,290 --> 00:37:50,670
Oh, actually in my list I
put them in different orders

545
00:37:50,670 --> 00:37:53,141
in the notes, but never mind.

546
00:37:53,141 --> 00:37:53,640
5.

547
00:37:56,560 --> 00:38:00,540
There's an additive inverse
in the vector space.

548
00:38:00,540 --> 00:38:07,230
So for each v belonging
to the vector space,

549
00:38:07,230 --> 00:38:17,310
there is a u belonging
to the vector space such

550
00:38:17,310 --> 00:38:25,000
that v plus u is equal to 0.

551
00:38:25,000 --> 00:38:31,250
So additive identity
you can find

552
00:38:31,250 --> 00:38:34,780
for every element
its opposite vector.

553
00:38:34,780 --> 00:38:36,150
It always can be found.

554
00:38:39,670 --> 00:38:44,240
And last is this
[INAUDIBLE] which

555
00:38:44,240 --> 00:38:52,820
says that a times u plus
v is equal to au plus av,

556
00:38:52,820 --> 00:39:01,660
and a plus b on v is
equal to av plus bv.

557
00:39:01,660 --> 00:39:06,220
And a's and b's
belong to the numbers.

558
00:39:06,220 --> 00:39:08,760
a and b's belong to the field.

559
00:39:08,760 --> 00:39:14,675
And u and v belong
to the vector space.

560
00:39:14,675 --> 00:39:15,175
OK.

561
00:39:18,180 --> 00:39:20,450
It's a little disconcerting.

562
00:39:20,450 --> 00:39:21,820
There's a lot of things.

563
00:39:21,820 --> 00:39:26,630
But actually, they
are quite minimal.

564
00:39:26,630 --> 00:39:28,422
It's well done, this definition.

565
00:39:28,422 --> 00:39:29,880
They're all kind
of things that you

566
00:39:29,880 --> 00:39:36,930
know that follow quite
immediately by little proofs.

567
00:39:36,930 --> 00:39:38,800
You will see more in
the notes, but let

568
00:39:38,800 --> 00:39:42,030
me just say briefly
a few of them.

569
00:39:42,030 --> 00:39:48,740
So here is the additive
identity, the vector 0.

570
00:39:48,740 --> 00:39:53,880
It's easy to prove that
this vector 0 is unique.

571
00:39:53,880 --> 00:39:58,890
If you find another 0 prime that
also satisfies this property,

572
00:39:58,890 --> 00:40:00,650
0 is equal to 0 prime.

573
00:40:00,650 --> 00:40:03,690
So it's unique.

574
00:40:03,690 --> 00:40:16,570
You can also show that 0 times
any vector is equal to 0.

575
00:40:16,570 --> 00:40:20,320
And here, this 0
belongs to the field

576
00:40:20,320 --> 00:40:23,250
and this 0 belongs
to the vector space.

577
00:40:23,250 --> 00:40:27,340
So the 0-- you had to postulate
that the 1 in the field

578
00:40:27,340 --> 00:40:29,710
does the right thing, but
you don't need to postulate

579
00:40:29,710 --> 00:40:33,730
that 0, the number 0,
multiplied by a vector is 0.

580
00:40:33,730 --> 00:40:35,480
You can prove that.

581
00:40:35,480 --> 00:40:37,930
And these are not
difficult to prove.

582
00:40:37,930 --> 00:40:41,210
All of them are
one-line exercises.

583
00:40:41,210 --> 00:40:43,110
They're done in that book.

584
00:40:43,110 --> 00:40:46,060
You can look at them.

585
00:40:46,060 --> 00:40:48,520
Moreover, another one.

586
00:40:48,520 --> 00:40:56,940
a any number times the 0 vector
is equal to the 0 vector.

587
00:40:56,940 --> 00:40:59,810
So in this case, those
both are vectors.

588
00:40:59,810 --> 00:41:03,690
That's also another property.

589
00:41:03,690 --> 00:41:08,740
So the 0 vector and the 0 number
really do the right thing.

590
00:41:08,740 --> 00:41:12,960
Then, another property,
the additive inverse.

591
00:41:12,960 --> 00:41:14,560
This is sort of interesting.

592
00:41:14,560 --> 00:41:21,240
So the additive inverse,
you can prove it's unique.

593
00:41:21,240 --> 00:41:22,960
So the additive
inverse is unique.

594
00:41:31,420 --> 00:41:44,040
And it's called-- for v, it's
called minus v, just a name.

595
00:41:44,040 --> 00:41:49,420
And actually, you can prove
it's equal to the number minus 1

596
00:41:49,420 --> 00:41:50,185
times the vector.

597
00:41:54,390 --> 00:41:59,380
Might sound totally trivial
but try to prove them.

598
00:41:59,380 --> 00:42:02,950
They're all simple, but they're
not trivial, all these things.

599
00:42:02,950 --> 00:42:08,200
So you call it minus v, but
it's actually-- this is a proof.

600
00:42:12,420 --> 00:42:14,170
OK.

601
00:42:14,170 --> 00:42:16,910
So examples.

602
00:42:16,910 --> 00:42:21,010
Let's do a few examples.

603
00:42:21,010 --> 00:42:24,125
I'll have five examples
that we're going to use.

604
00:42:28,330 --> 00:42:35,180
So I think the main thing for
a physicist that I remember

605
00:42:35,180 --> 00:42:37,930
being confused about
is the statement

606
00:42:37,930 --> 00:42:41,500
that there's no characterization
that the vectors are

607
00:42:41,500 --> 00:42:42,930
real or complex.

608
00:42:42,930 --> 00:42:45,240
The vectors are
the vectors and you

609
00:42:45,240 --> 00:42:47,690
multiply by a real
or complex numbers.

610
00:42:47,690 --> 00:42:51,293
So I will have one example
that makes that very dramatic.

611
00:42:54,520 --> 00:42:57,320
As dramatic as it can be.

612
00:42:57,320 --> 00:43:10,280
So one example of vector spaces,
the set of N component vectors.

613
00:43:10,280 --> 00:43:16,055
So here it is,
a1, a2, up to a n.

614
00:43:16,055 --> 00:43:23,030
For example, with capital N.
With a i belongs to the real

615
00:43:23,030 --> 00:43:35,140
and i going from 1 up
to N is a vector space

616
00:43:35,140 --> 00:43:41,330
over r, the real numbers.

617
00:43:41,330 --> 00:43:45,930
So people use that
terminology, a vector space

618
00:43:45,930 --> 00:43:49,240
over the kind of numbers.

619
00:43:49,240 --> 00:43:51,270
You could call it
also a real vector

620
00:43:51,270 --> 00:43:52,920
space, that would be the same.

621
00:43:52,920 --> 00:43:55,490
You see, these
components are real.

622
00:43:55,490 --> 00:43:58,430
And you have to
think for a second

623
00:43:58,430 --> 00:44:02,020
if you believe all of them are
true or how would you do it.

624
00:44:02,020 --> 00:44:05,330
Well, if I would
be really precise,

625
00:44:05,330 --> 00:44:07,060
I would have to tell
you a lot of things

626
00:44:07,060 --> 00:44:08,370
that you would find boring.

627
00:44:08,370 --> 00:44:12,430
That, for example, you have
this vector and you add a set

628
00:44:12,430 --> 00:44:13,120
of b's.

629
00:44:13,120 --> 00:44:14,990
Well, you add the components.

630
00:44:14,990 --> 00:44:17,140
That's the definition of plus.

631
00:44:17,140 --> 00:44:19,930
And what's the definition
of multiplying by a number?

632
00:44:19,930 --> 00:44:22,860
Well, if a number is
multiplied by this vector,

633
00:44:22,860 --> 00:44:25,950
it goes in and
multiplies everybody.

634
00:44:25,950 --> 00:44:29,160
Those are implicit, or you
can fill-in the details.

635
00:44:29,160 --> 00:44:31,110
But if you define
them that way, it

636
00:44:31,110 --> 00:44:33,340
will satisfy all the properties.

637
00:44:33,340 --> 00:44:35,040
What is the 0 vector?

638
00:44:35,040 --> 00:44:39,220
It must be the one
with all entries 0.

639
00:44:39,220 --> 00:44:41,360
What is the additive inverse?

640
00:44:41,360 --> 00:44:43,920
Well, change the sign
of all these things.

641
00:44:43,920 --> 00:44:48,380
So it's kind of obvious that
this satisfies everything,

642
00:44:48,380 --> 00:44:52,250
if you understand how the sum
and the multiplication goes.

643
00:44:54,970 --> 00:44:57,970
Another one, it's
kind of similar.

644
00:44:57,970 --> 00:44:59,800
2.

645
00:44:59,800 --> 00:45:10,625
The set of M cross N matrices
with complex entries.

646
00:45:13,966 --> 00:45:18,120
Complex entries.

647
00:45:18,120 --> 00:45:24,400
So here you have it,
a1 1, a1 2, a1 N.

648
00:45:24,400 --> 00:45:30,330
And here it goes up
to aM1, aM2, aMN.

649
00:45:35,560 --> 00:45:45,630
With all the a i j's belonging
to the complex numbers,

650
00:45:45,630 --> 00:45:52,540
then-- I'll erase here.

651
00:45:52,540 --> 00:45:56,300
Then you have that this
is a complex vector space.

652
00:46:00,590 --> 00:46:09,415
Is a complex vector space.

653
00:46:12,760 --> 00:46:14,710
How do you multiply by a number?

654
00:46:14,710 --> 00:46:18,070
You multiply a number times
every entry of the matrices.

655
00:46:18,070 --> 00:46:20,590
How do sum two matrices?

656
00:46:20,590 --> 00:46:25,000
They have the same size, so
you sum each element the way

657
00:46:25,000 --> 00:46:25,740
it should be.

658
00:46:25,740 --> 00:46:28,860
And that should
be a vector space.

659
00:46:31,630 --> 00:46:34,100
Here is an example
that is, perhaps,

660
00:46:34,100 --> 00:46:37,070
a little more surprising.

661
00:46:37,070 --> 00:46:54,150
So the space of 2 by
2 Hermitian matrices

662
00:46:54,150 --> 00:46:58,415
is a real vector space.

663
00:47:06,450 --> 00:47:11,050
You see, this can be easily
thought [INAUDIBLE] naturally

664
00:47:11,050 --> 00:47:12,600
thought as a real vector space.

665
00:47:12,600 --> 00:47:16,750
This is a little surprising
because Hermitian matrices have

666
00:47:16,750 --> 00:47:17,640
i's.

667
00:47:17,640 --> 00:47:21,370
You remember the most
general Hermitian matrix

668
00:47:21,370 --> 00:47:30,470
was of the form--
well, a plus-- no,

669
00:47:30,470 --> 00:47:38,790
c plus d, c minus d,
a plus ib, a minus ib,

670
00:47:38,790 --> 00:47:44,480
with all these numbers
c, d, b in real.

671
00:47:44,480 --> 00:47:47,870
But they're complex numbers.

672
00:47:47,870 --> 00:47:51,960
Why is this naturally
a real vector space?

673
00:47:51,960 --> 00:47:56,590
The problem is that if
you multiply by a number,

674
00:47:56,590 --> 00:47:59,530
it should still be a
Hermitian matrix in order

675
00:47:59,530 --> 00:48:01,825
for it to be a vector space.

676
00:48:01,825 --> 00:48:03,250
It should be in the vector.

677
00:48:03,250 --> 00:48:06,430
But if you multiply by a real
number, there's no problem.

678
00:48:06,430 --> 00:48:08,670
The matrix remains Hermitian.

679
00:48:08,670 --> 00:48:10,640
You multiplied by
a complex number,

680
00:48:10,640 --> 00:48:12,550
you use the Hermiticity.

681
00:48:12,550 --> 00:48:16,440
But an i somewhere here
for all the factors and it

682
00:48:16,440 --> 00:48:18,790
will not be Hermitian.

683
00:48:18,790 --> 00:48:22,620
So this is why it's
a real vector space.

684
00:48:22,620 --> 00:48:31,180
Multiplication by real
numbers preserves Hermiticity.

685
00:48:38,400 --> 00:48:41,620
So that's surprising.

686
00:48:41,620 --> 00:48:44,520
So again, illustrates
that nobody

687
00:48:44,520 --> 00:48:47,170
would say this is a real vector.

688
00:48:47,170 --> 00:48:52,130
But it really should be thought
as a vector over real numbers.

689
00:48:52,130 --> 00:48:56,130
Vector space over real numbers.

690
00:48:56,130 --> 00:48:58,640
Two more examples.

691
00:48:58,640 --> 00:49:04,250
And they are kind
of interesting.

692
00:49:16,210 --> 00:49:22,710
So the next example is the set
of polynomials as vector space.

693
00:49:22,710 --> 00:49:26,210
So that, again, is sort of
a very imaginative thing.

694
00:49:26,210 --> 00:49:33,600
The set of polynomials p of z.

695
00:49:37,380 --> 00:49:45,230
Here, z belongs to some
field and p of z, which

696
00:49:45,230 --> 00:49:50,120
is a function of z, also
belongs to the same field.

697
00:49:50,120 --> 00:49:52,700
And each polynomial
has coefficient.

698
00:49:52,700 --> 00:50:01,790
So any p of z is a0
plus a1 z plus a2 z

699
00:50:01,790 --> 00:50:07,170
squared plus-- up to some an zn.

700
00:50:07,170 --> 00:50:10,580
A polynomial is
supposed to end That's

701
00:50:10,580 --> 00:50:12,160
pretty important
about polynomials.

702
00:50:12,160 --> 00:50:16,330
So the dots don't go up forever.

703
00:50:16,330 --> 00:50:21,770
So here it is, the a i's
also belong to the field.

704
00:50:21,770 --> 00:50:23,030
So looked at this polynomials.

705
00:50:26,310 --> 00:50:29,630
We have the letter z and
they have these coefficients

706
00:50:29,630 --> 00:50:30,520
which are numbers.

707
00:50:30,520 --> 00:50:38,230
So a real polynomial-- you
know 2 plus x plus x squared.

708
00:50:38,230 --> 00:50:42,380
So you have your real numbers
times this general variable

709
00:50:42,380 --> 00:50:44,910
that it's also
supposed to be real.

710
00:50:44,910 --> 00:50:47,250
So you could have it real.

711
00:50:47,250 --> 00:50:48,370
You could have it complex.

712
00:50:48,370 --> 00:50:50,540
So that's a polynomial.

713
00:50:50,540 --> 00:50:53,170
How is that a vector space?

714
00:50:53,170 --> 00:51:00,190
Well, it's a vector
space-- the space

715
00:51:00,190 --> 00:51:16,490
p of F of those polynomials--
of all polynomials

716
00:51:16,490 --> 00:51:25,410
is a vector space over
F. And why is that?

717
00:51:25,410 --> 00:51:27,910
Well, you can take--
again, there's

718
00:51:27,910 --> 00:51:29,620
some implicit definitions.

719
00:51:29,620 --> 00:51:31,410
How do you sum polynomials?

720
00:51:31,410 --> 00:51:34,540
Well, you sum the
independent coefficients.

721
00:51:34,540 --> 00:51:37,440
You just sum them
and factor out.

722
00:51:37,440 --> 00:51:40,910
So there's an obvious
definition of sum.

723
00:51:40,910 --> 00:51:44,120
How do you multiply a
polynomial by a number?

724
00:51:44,120 --> 00:51:47,550
Obvious definition, you
multiply everything by a number.

725
00:51:47,550 --> 00:51:50,150
If you sum polynomials,
you get polynomials.

726
00:51:50,150 --> 00:51:53,690
Given a polynomial, there
is a negative polynomial

727
00:51:53,690 --> 00:51:56,410
that adds up to 0.

728
00:51:56,410 --> 00:52:00,480
There's a 0 when all
the coefficients is 0.

729
00:52:00,480 --> 00:52:02,760
And it has all the
nice properties.

730
00:52:02,760 --> 00:52:06,710
Now, this example
is more nontrivial

731
00:52:06,710 --> 00:52:10,710
because you would
think, as opposed

732
00:52:10,710 --> 00:52:13,830
to the previous examples,
that this is probably

733
00:52:13,830 --> 00:52:17,960
infinite dimensional because
it has the linear polynomial,

734
00:52:17,960 --> 00:52:21,360
the quadratic, the cubic,
the quartic, the quintic, all

735
00:52:21,360 --> 00:52:23,500
of them together.

736
00:52:23,500 --> 00:52:28,090
And yes, we'll see
that in a second.

737
00:52:28,090 --> 00:52:31,900
So set of polynomials.

738
00:52:31,900 --> 00:52:32,740
5.

739
00:52:32,740 --> 00:52:34,210
Another example, 5.

740
00:52:37,200 --> 00:52:43,660
The set F infinity of
infinite sequences.

741
00:52:49,680 --> 00:52:55,800
Sequences x1, x2, infinite
sequences where the x i's

742
00:52:55,800 --> 00:52:59,170
are in the field.

743
00:52:59,170 --> 00:53:01,210
So you've got an
infinite sequence

744
00:53:01,210 --> 00:53:03,610
and you want to add
another infinite sequence.

745
00:53:03,610 --> 00:53:05,890
Well, you add the first
element, the second elements.

746
00:53:05,890 --> 00:53:08,433
It's like an infinite
column vector.

747
00:53:08,433 --> 00:53:12,900
Sometimes mathematicians like to
write column vectors like that

748
00:53:12,900 --> 00:53:14,460
because it's practical.

749
00:53:14,460 --> 00:53:16,680
It saves space on a page.

750
00:53:16,680 --> 00:53:21,010
The vertical one, you
start writing and the pages

751
00:53:21,010 --> 00:53:22,020
grow very fast.

752
00:53:22,020 --> 00:53:24,840
So here's an infinite sequence.

753
00:53:24,840 --> 00:53:28,330
And think of it as a
vertical one if you wish.

754
00:53:28,330 --> 00:53:30,350
And all elements
are here, but there

755
00:53:30,350 --> 00:53:34,310
are infinitely many
in every sequence.

756
00:53:34,310 --> 00:53:40,140
And of course, the set of all
infinite sequences is infinite.

757
00:53:40,140 --> 00:53:43,380
So this is a vector
space over F.

758
00:53:43,380 --> 00:53:45,580
Again, because all
the numbers are here,

759
00:53:45,580 --> 00:53:55,820
so it's a vector space over F.

760
00:53:55,820 --> 00:53:57,400
And last example.

761
00:54:07,340 --> 00:54:12,470
Our last example is a
familiar one in physics,

762
00:54:12,470 --> 00:54:18,010
is the set of complex
functions in an interval.

763
00:54:18,010 --> 00:54:36,110
Set of complex functions on
an interval x from 0 to L.

764
00:54:36,110 --> 00:54:39,490
So a set of complex
functions f of x

765
00:54:39,490 --> 00:54:41,930
I could put here on an
interval [INAUDIBLE].

766
00:54:41,930 --> 00:54:47,810
So this is a complex
vector space.

767
00:54:47,810 --> 00:54:49,060
Vector space.

768
00:54:54,290 --> 00:54:58,080
The last three
examples, probably you

769
00:54:58,080 --> 00:55:01,570
would agree that there
are infinite dimensional,

770
00:55:01,570 --> 00:55:06,920
even though I've not defined
what that means very precisely.

771
00:55:06,920 --> 00:55:09,340
But that's what we're going
to try to understand now.

772
00:55:09,340 --> 00:55:12,490
We're supposed to understand
the concept of dimensionality.

773
00:55:12,490 --> 00:55:16,580
So let's get to
that concept now.

774
00:55:16,580 --> 00:55:23,010
So in terms of dimensionality,
to build this idea

775
00:55:23,010 --> 00:55:24,743
you need a definition.

776
00:55:27,830 --> 00:55:32,670
You need to know the term
subspace of a vector space.

777
00:55:32,670 --> 00:55:36,160
What is a subspace
of a vector space?

778
00:55:36,160 --> 00:55:41,320
A subspace of a vector space
is a subset of the vector space

779
00:55:41,320 --> 00:55:43,400
that is still a vector space.

780
00:55:43,400 --> 00:55:45,920
So that's why it's
called subspace.

781
00:55:45,920 --> 00:55:47,560
It's different from subset.

782
00:55:47,560 --> 00:56:08,870
So a subspace of V is a subset
of V that is a vector space.

783
00:56:14,690 --> 00:56:18,800
So in particular, it
must contain the vector 0

784
00:56:18,800 --> 00:56:23,535
because any vector space
contains the vector 0.

785
00:56:27,410 --> 00:56:31,060
One of the ways you sometimes
want to understand the vector

786
00:56:31,060 --> 00:56:37,500
space is by representing it as
a sum of smaller vector spaces.

787
00:56:37,500 --> 00:56:40,180
And we will do that when
we consider, for example,

788
00:56:40,180 --> 00:56:42,160
angular momentum in detail.

789
00:56:42,160 --> 00:56:51,770
So you want to write a vector
space as a sum of subspaces.

790
00:56:51,770 --> 00:56:53,780
So what is that called?

791
00:56:53,780 --> 00:56:56,220
It's called a direct sum.

792
00:56:56,220 --> 00:57:02,400
So if you can write--
here is the equation.

793
00:57:02,400 --> 00:57:09,570
You say V is equal to u1
direct sum with u2 direct sum

794
00:57:09,570 --> 00:57:16,680
with u3 direct sum with u m.

795
00:57:16,680 --> 00:57:23,970
When we say this, we
mean the following.

796
00:57:23,970 --> 00:57:33,276
That the ui's are
subspaces of V.

797
00:57:33,276 --> 00:57:42,760
And any V in the
vector space can

798
00:57:42,760 --> 00:58:01,550
be written uniquely as a1
u1 plus a2 u2 plus a n u

799
00:58:01,550 --> 00:58:10,130
n with ui [INAUDIBLE]
capital Ui.

800
00:58:10,130 --> 00:58:14,500
So let me review
what we just said.

801
00:58:14,500 --> 00:58:17,190
So you have a
vector space and you

802
00:58:17,190 --> 00:58:21,750
want to decompose it in
sort of basic ingredients.

803
00:58:21,750 --> 00:58:23,680
This is called a direct sum.

804
00:58:26,460 --> 00:58:30,270
V is a direct sum of subspaces.

805
00:58:30,270 --> 00:58:31,237
Direct sum.

806
00:58:35,140 --> 00:58:39,300
And the Ui's are
subspaces of V. But what

807
00:58:39,300 --> 00:58:42,090
must happen for
this to be true is

808
00:58:42,090 --> 00:58:45,310
that once you take
any vector here,

809
00:58:45,310 --> 00:58:47,800
you can write it as
a sum of a vector

810
00:58:47,800 --> 00:58:51,530
here, a vector here, a vector
here, a vector everywhere.

811
00:58:51,530 --> 00:58:54,040
And it must be done uniquely.

812
00:58:54,040 --> 00:58:56,810
If you can do this
in more than one way,

813
00:58:56,810 --> 00:58:59,160
this is not a direct sum.

814
00:58:59,160 --> 00:59:02,270
These subspaces kind of overlap.

815
00:59:02,270 --> 00:59:06,056
They're not doing the
decomposition in a minimal way.

816
00:59:06,056 --> 00:59:06,760
Yes.

817
00:59:06,760 --> 00:59:09,010
AUDIENCE: Does the expression
of V have to be a linear

818
00:59:09,010 --> 00:59:10,600
combination of the
vectors of the U,

819
00:59:10,600 --> 00:59:15,367
or just sums of the U sub i's?

820
00:59:15,367 --> 00:59:17,033
PROFESSOR: It's some
linear combination.

821
00:59:21,430 --> 00:59:23,900
Look, the interpretation,
for example, R2.

822
00:59:26,670 --> 00:59:29,880
The normal vector space R2.

823
00:59:29,880 --> 00:59:34,100
You have an intuition quite
clearly that any vector here

824
00:59:34,100 --> 00:59:41,150
is a unique sum of this
component along this subspace

825
00:59:41,150 --> 00:59:45,220
and this component
along this subspace.

826
00:59:45,220 --> 00:59:52,730
So it's a trivial example,
but the vector space R2

827
00:59:52,730 --> 00:59:58,120
has a vector subspace R1 here
and a vector subspace R1.

828
00:59:58,120 --> 01:00:01,250
Any vector in R2
is uniquely written

829
01:00:01,250 --> 01:00:03,780
as a sum of these two vectors.

830
01:00:03,780 --> 01:00:09,345
That means that R2
is really R1 plus R1.

831
01:00:12,020 --> 01:00:13,093
Yes.

832
01:00:13,093 --> 01:00:14,905
AUDIENCE: [INAUDIBLE].

833
01:00:14,905 --> 01:00:19,520
Is it redundant to say that
that-- because a1 u1 is also

834
01:00:19,520 --> 01:00:22,030
in big U sub 1.

835
01:00:22,030 --> 01:00:22,900
PROFESSOR: Oh.

836
01:00:22,900 --> 01:00:23,590
Oh, yes.

837
01:00:23,590 --> 01:00:24,430
You're right.

838
01:00:24,430 --> 01:00:25,460
No, I'm sorry.

839
01:00:25,460 --> 01:00:26,520
I shouldn't write those.

840
01:00:26,520 --> 01:00:28,720
I'm sorry.

841
01:00:28,720 --> 01:00:31,040
That's absolutely right.

842
01:00:31,040 --> 01:00:34,770
If I had that in my
notes, it was a mistake.

843
01:00:34,770 --> 01:00:35,370
Thank you.

844
01:00:35,370 --> 01:00:36,380
That was very good.

845
01:00:36,380 --> 01:00:38,470
Did I have that in my notes?

846
01:00:38,470 --> 01:00:42,030
No, I had it as you said it.

847
01:00:42,030 --> 01:00:42,560
True.

848
01:00:42,560 --> 01:00:46,770
So can be written
uniquely as a vector in

849
01:00:46,770 --> 01:00:48,310
first, a vector in the second.

850
01:00:48,310 --> 01:00:52,220
And the a's are
absolutely not necessary.

851
01:00:52,220 --> 01:00:53,200
OK.

852
01:00:53,200 --> 01:01:06,070
So let's go ahead then and
say the following things.

853
01:01:06,070 --> 01:01:07,870
So here we're
going to try to get

854
01:01:07,870 --> 01:01:14,210
to the concept of
dimensionality in a precise way.

855
01:01:14,210 --> 01:01:16,269
Yes.

856
01:01:16,269 --> 01:01:18,684
AUDIENCE: [INAUDIBLE].

857
01:01:18,684 --> 01:01:21,082
PROFESSOR: Right,
the last one is m.

858
01:01:21,082 --> 01:01:21,582
Thank you.

859
01:01:38,600 --> 01:01:39,290
All right.

860
01:01:46,220 --> 01:01:49,070
The concept of dimensionality
of a vector space

861
01:01:49,070 --> 01:01:51,350
is something that you
intuitively understand.

862
01:01:51,350 --> 01:01:56,900
It's sort of how many
linearly independent vectors

863
01:01:56,900 --> 01:02:01,610
you need to describe the
whole set of vectors.

864
01:02:01,610 --> 01:02:05,970
So that is the number
you're trying to get to.

865
01:02:05,970 --> 01:02:10,320
I'll follow it up in a
slightly rigorous way

866
01:02:10,320 --> 01:02:13,150
to be able to do infinite
dimensional space as well.

867
01:02:13,150 --> 01:02:18,470
So we will consider something
called a list of vectors.

868
01:02:18,470 --> 01:02:22,210
List of vectors.

869
01:02:22,210 --> 01:02:27,020
And that will be something
like v1, v2 vectors in a vector

870
01:02:27,020 --> 01:02:28,050
space up to vn.

871
01:02:30,640 --> 01:02:38,650
Any list of vectors
has finite length.

872
01:02:38,650 --> 01:02:44,390
So we don't accept infinite
lists by definition.

873
01:02:48,480 --> 01:02:51,420
You can ask, once you
have a list of vectors,

874
01:02:51,420 --> 01:02:57,580
what is the vector subspace
spanned by this list?

875
01:02:57,580 --> 01:03:00,420
How much do you
reach with that list?

876
01:03:00,420 --> 01:03:04,440
So we call it the
span of the list.

877
01:03:04,440 --> 01:03:10,680
The span of the list, vn.

878
01:03:10,680 --> 01:03:23,570
And it's the set of all linear
combinations a1 v1 plus a2 v2

879
01:03:23,570 --> 01:03:32,500
plus a n vn for ai in the field.

880
01:03:32,500 --> 01:03:36,200
So the span of the list
is all possible products

881
01:03:36,200 --> 01:03:42,915
of your vectors on the list
are-- and put like that.

882
01:03:42,915 --> 01:03:51,620
So if we say that the
list spans a vector space,

883
01:03:51,620 --> 01:03:55,170
if the span of the list
is the vector space.

884
01:03:55,170 --> 01:03:57,650
So that's natural language.

885
01:03:57,650 --> 01:04:01,010
We say, OK, this list
spans the vector space.

886
01:04:01,010 --> 01:04:01,520
Why?

887
01:04:01,520 --> 01:04:04,170
Because if you produce
the span of the list,

888
01:04:04,170 --> 01:04:06,820
it fills a vector space.

889
01:04:06,820 --> 01:04:12,050
OK, so I could say it that way.

890
01:04:12,050 --> 01:04:26,710
So here is the definition,
V is finite dimensional

891
01:04:26,710 --> 01:04:29,180
if it's spanned by some list.

892
01:04:29,180 --> 01:04:35,480
If V is spanned by some list.

893
01:04:40,180 --> 01:04:41,620
So why is that?

894
01:04:41,620 --> 01:04:47,200
Because if the list is-- a
definition, finite dimensional.

895
01:04:47,200 --> 01:04:48,800
If it's spanned by some list.

896
01:04:48,800 --> 01:04:52,090
If you got your list, by
definition it's finite length.

897
01:04:52,090 --> 01:04:54,790
And with some set of
vectors, you span everything.

898
01:04:58,380 --> 01:05:02,990
And moreover, it's
infinite dimensional

899
01:05:02,990 --> 01:05:05,840
if it's not finite dimensional.

900
01:05:05,840 --> 01:05:10,775
It's kind of silly,
but infinite-- a space

901
01:05:10,775 --> 01:05:24,280
V is infinite dimensional if
it is not finite dimensional.

902
01:05:24,280 --> 01:05:32,370
Which is to say that there is
no list that spans the space.

903
01:05:32,370 --> 01:05:36,210
So for example, this definition
is tailored in a nice way.

904
01:05:36,210 --> 01:05:38,116
Like let's think
of the polynomials.

905
01:05:42,010 --> 01:05:45,270
And we want to see if it's
finite dimensional or infinite

906
01:05:45,270 --> 01:05:45,980
dimensional.

907
01:05:45,980 --> 01:05:51,160
So you claim it's
finite dimensional.

908
01:05:51,160 --> 01:05:53,020
Let's see if it's
finite dimensional.

909
01:05:53,020 --> 01:05:55,800
So we make a list
of polynomials.

910
01:05:55,800 --> 01:06:00,630
The list must have some length,
at least, that spans it.

911
01:06:00,630 --> 01:06:03,800
You put all these
730 polynomials

912
01:06:03,800 --> 01:06:09,830
that you think span the list,
span the space, in this list.

913
01:06:09,830 --> 01:06:12,900
Now, if you look at
the list, it's 720.

914
01:06:12,900 --> 01:06:14,800
You can check one
by one until you

915
01:06:14,800 --> 01:06:17,385
find what is the one
of highest order,

916
01:06:17,385 --> 01:06:20,700
the polynomial of
highest degree.

917
01:06:20,700 --> 01:06:26,560
But if the highest degree
is say, z to the 1 million,

918
01:06:26,560 --> 01:06:30,540
then any polynomial that has
a z to the 2 million cannot be

919
01:06:30,540 --> 01:06:32,110
spanned by this one.

920
01:06:32,110 --> 01:06:36,240
So there's no finite
list that can span this,

921
01:06:36,240 --> 01:06:40,820
so this set-- the
example in 4 is

922
01:06:40,820 --> 01:06:42,720
infinite dimensional for sure.

923
01:06:45,240 --> 01:06:49,590
Example 4 is
infinite dimensional.

924
01:06:57,060 --> 01:07:04,940
Well, example one is
finite dimensional.

925
01:07:07,560 --> 01:07:09,900
You can see that
because we can produce

926
01:07:09,900 --> 01:07:13,250
a list that spans the space.

927
01:07:13,250 --> 01:07:14,965
So look at the example 1.

928
01:07:17,620 --> 01:07:18,120
It's there.

929
01:07:22,710 --> 01:07:24,555
Well, what would be the list?

930
01:07:24,555 --> 01:07:28,090
The list would be-- list.

931
01:07:28,090 --> 01:07:33,310
You would put a vector
e1, e2, up to en.

932
01:07:33,310 --> 01:07:40,500
And the vector e1
would be 1, 0, 0, 0, 0.

933
01:07:40,500 --> 01:07:45,680
The vector e2 would
be 0, 1, 0, 0, 0.

934
01:07:45,680 --> 01:07:47,300
And go on like that.

935
01:07:47,300 --> 01:07:50,190
So you put 1's and 0's.

936
01:07:50,190 --> 01:07:52,000
And you have n of them.

937
01:07:52,000 --> 01:07:58,540
And certainly, the most general
one is a1 times e1 a2 times e2.

938
01:07:58,540 --> 01:08:00,550
And you got the list.

939
01:08:00,550 --> 01:08:06,160
So example 1 is
finite dimensional.

940
01:08:06,160 --> 01:08:10,320
A list of vectors is
linearly independent.

941
01:08:10,320 --> 01:08:23,250
A list is linearly independent
if a list v1 up to vn

942
01:08:23,250 --> 01:08:32,649
is linearly independent, If
a1 v1 plus a2 v2 plus a n vn

943
01:08:32,649 --> 01:08:44,710
is equal to 0 has the unique
solution a1 equal a2 equal

944
01:08:44,710 --> 01:08:48,649
all of them equal 0.

945
01:08:48,649 --> 01:08:57,990
So that is to mean that
whenever this list satisfies

946
01:08:57,990 --> 01:09:02,060
this property-- if you want
to represent the vector

947
01:09:02,060 --> 01:09:05,859
0 with this list, you
must set all of them

948
01:09:05,859 --> 01:09:08,260
equal to 0, all
the coefficients.

949
01:09:08,260 --> 01:09:10,850
That's clear as well
in this example.

950
01:09:10,850 --> 01:09:13,170
If you want to
represent the 0 vector,

951
01:09:13,170 --> 01:09:18,370
you must have 0 component
against the basis vector x

952
01:09:18,370 --> 01:09:19,920
and basis vector y.

953
01:09:19,920 --> 01:09:23,550
So the list of this
vector and this vector

954
01:09:23,550 --> 01:09:27,069
is linearly independent
because the 0 vector

955
01:09:27,069 --> 01:09:31,180
must have 0 numbers
multiplying each of them.

956
01:09:31,180 --> 01:09:36,830
So finally, we define
what is a basis.

957
01:09:36,830 --> 01:09:58,266
A basis of V is a
list of vectors in V

958
01:09:58,266 --> 01:10:07,255
that spans V and is
linearly independent.

959
01:10:13,500 --> 01:10:16,200
So what is a basis?

960
01:10:16,200 --> 01:10:18,420
Well, you should
have enough vectors

961
01:10:18,420 --> 01:10:21,970
to represent every vector.

962
01:10:21,970 --> 01:10:25,970
So it must span V. And
what else should it have?

963
01:10:25,970 --> 01:10:29,650
It shouldn't have extra
vectors that you don't need.

964
01:10:29,650 --> 01:10:31,110
It should be minimal.

965
01:10:31,110 --> 01:10:33,150
It should be all
linearly independent.

966
01:10:33,150 --> 01:10:36,610
You shouldn't have
added more stuff to it.

967
01:10:36,610 --> 01:10:43,040
So any finite dimensional
vector space has a basis.

968
01:10:45,740 --> 01:10:50,200
It's easy to do it.

969
01:10:50,200 --> 01:10:53,170
There's another thing
that one can prove.

970
01:10:53,170 --> 01:10:58,110
It may look kind of obvious,
but it requires a small proof

971
01:10:58,110 --> 01:11:00,940
that if you have-- the
bases are not unique.

972
01:11:00,940 --> 01:11:03,070
It's something we're going
to exploit all the time.

973
01:11:03,070 --> 01:11:05,800
One basis, another
basis, a third basis.

974
01:11:05,800 --> 01:11:08,650
We're going to change
basis all the time.

975
01:11:08,650 --> 01:11:13,070
Well, the bases are not
unique, but the length

976
01:11:13,070 --> 01:11:17,130
of the bases of a vector
space is always the same.

977
01:11:17,130 --> 01:11:20,850
So the length of the
list is-- a number

978
01:11:20,850 --> 01:11:23,810
is the same whatever
base you choose.

979
01:11:23,810 --> 01:11:25,920
And that length
is what is called

980
01:11:25,920 --> 01:11:28,610
the dimension of
the vector space.

981
01:11:28,610 --> 01:11:41,700
So the dimension
of a vector space

982
01:11:41,700 --> 01:12:00,230
is the length of any
bases of V. And therefore,

983
01:12:00,230 --> 01:12:01,850
it's a well-defined concept.

984
01:12:01,850 --> 01:12:06,060
Any base of a finite vector
space has the same length,

985
01:12:06,060 --> 01:12:08,860
and the dimension
is that number.

986
01:12:08,860 --> 01:12:11,070
So there was a question.

987
01:12:11,070 --> 01:12:12,658
Yes?

988
01:12:12,658 --> 01:12:14,514
AUDIENCE: Is there
any difference

989
01:12:14,514 --> 01:12:16,370
between bases [INAUDIBLE]?

990
01:12:20,900 --> 01:12:22,900
PROFESSOR: No, absolutely not.

991
01:12:22,900 --> 01:12:25,600
You could have a
basis, for example,

992
01:12:25,600 --> 01:12:28,570
of R2, which is this vector.

993
01:12:28,570 --> 01:12:33,010
The first and the
second is this vector.

994
01:12:33,010 --> 01:12:37,020
And any vector is a
linear superposition

995
01:12:37,020 --> 01:12:40,680
of these two vectors with some
coefficients and it's unique.

996
01:12:40,680 --> 01:12:43,580
You can find the coefficients.

997
01:12:43,580 --> 01:12:46,426
AUDIENCE: [INAUDIBLE].

998
01:12:46,426 --> 01:12:47,050
PROFESSOR: Yes.

999
01:12:47,050 --> 01:12:52,140
But you see, here is exactly
what I wanted to make clear.

1000
01:12:52,140 --> 01:12:54,480
We're putting the
vector space and we're

1001
01:12:54,480 --> 01:12:56,630
putting the least
possible structure.

1002
01:12:56,630 --> 01:13:00,650
I didn't say how to take the
inner product of two vectors.

1003
01:13:00,650 --> 01:13:02,660
It's not a definition
of a vector space.

1004
01:13:02,660 --> 01:13:04,610
It's something we'll put later.

1005
01:13:04,610 --> 01:13:08,190
And then, we will be able
to ask whether the basis is

1006
01:13:08,190 --> 01:13:09,790
orthonormal or not.

1007
01:13:09,790 --> 01:13:12,240
But the basis exists.

1008
01:13:12,240 --> 01:13:15,520
Even though you have no
definition of an inner product,

1009
01:13:15,520 --> 01:13:19,610
you can talk about basis
without any confusion.

1010
01:13:19,610 --> 01:13:23,420
You can also talk about
the matrix representation

1011
01:13:23,420 --> 01:13:24,530
of an operator.

1012
01:13:24,530 --> 01:13:27,230
And you don't need
an inner product,

1013
01:13:27,230 --> 01:13:30,370
which is sometimes very unclear.

1014
01:13:30,370 --> 01:13:34,490
You can talk about the
trace of an operator

1015
01:13:34,490 --> 01:13:37,390
and you don't need
an inner product.

1016
01:13:37,390 --> 01:13:40,610
You can talk about
eigenvectors and eigenvalues

1017
01:13:40,610 --> 01:13:43,470
and you don't need
an inner product.

1018
01:13:43,470 --> 01:13:45,185
The only thing you
need the inner product

1019
01:13:45,185 --> 01:13:46,820
is to get numbers.

1020
01:13:46,820 --> 01:13:51,330
And we'll use them to use
[INAUDIBLE] to get numbers.

1021
01:13:51,330 --> 01:13:53,380
But it can wait.

1022
01:13:53,380 --> 01:13:55,490
It's better than
you see all that you

1023
01:13:55,490 --> 01:13:58,770
can do without
introducing more things,

1024
01:13:58,770 --> 01:14:00,980
and then introduce them.

1025
01:14:00,980 --> 01:14:07,020
So let me explain a
little more this concept.

1026
01:14:07,020 --> 01:14:12,900
We were talking about this
base, this vector space 1,

1027
01:14:12,900 --> 01:14:13,590
for example.

1028
01:14:13,590 --> 01:14:20,645
And we produced a list that
spans e1, e2, up to en.

1029
01:14:20,645 --> 01:14:23,000
And those were these vectors.

1030
01:14:23,000 --> 01:14:26,340
Now, this list not
only spans, but they

1031
01:14:26,340 --> 01:14:28,010
are linearly independent.

1032
01:14:28,010 --> 01:14:31,650
If you put a1 times
this plus a2 times

1033
01:14:31,650 --> 01:14:33,520
this and you set
it all equal to 0.

1034
01:14:33,520 --> 01:14:37,990
Well, each entry will be
0, and all the a's are 0.

1035
01:14:37,990 --> 01:14:42,880
So these e's that you put
here on that list is actually

1036
01:14:42,880 --> 01:14:44,520
a basis.

1037
01:14:44,520 --> 01:14:47,810
Therefore, the length of that
basis is the dimensionality.

1038
01:14:47,810 --> 01:14:54,700
And this space has
dimension N. You

1039
01:14:54,700 --> 01:14:58,960
should be able to prove that
this space has been dimension

1040
01:14:58,960 --> 01:15:12,550
m times N. Now, let me do the
Hermitian-- these matrices.

1041
01:15:12,550 --> 01:15:15,670
And try to figure out
the dimensionality

1042
01:15:15,670 --> 01:15:19,360
of the space of
Hermitian matrices.

1043
01:15:19,360 --> 01:15:20,870
So here they are.

1044
01:15:20,870 --> 01:15:24,390
This is the most general
Hermitian matrix.

1045
01:15:24,390 --> 01:15:31,190
And I'm going to produce for
you a list of four vectors.

1046
01:15:31,190 --> 01:15:34,570
Vectors-- yes, they're matrices,
but we call them vectors.

1047
01:15:34,570 --> 01:15:35,900
So here is the list.

1048
01:15:40,340 --> 01:15:45,110
The unit matrix, the first
Pauli matrix, the second Pauli

1049
01:15:45,110 --> 01:15:49,270
matrix, and the
third Pauli matrix.

1050
01:15:49,270 --> 01:15:53,330
All right, let's see how
far do we get from there.

1051
01:15:53,330 --> 01:15:56,710
OK, this is a list of
vectors in the vector space

1052
01:15:56,710 --> 01:15:59,090
because all of
them are Hermitian.

1053
01:15:59,090 --> 01:15:59,590
Good.

1054
01:16:02,690 --> 01:16:04,250
Do they span?

1055
01:16:04,250 --> 01:16:08,570
Well, you calculate the most
general Hermitian matrix

1056
01:16:08,570 --> 01:16:09,420
of this form.

1057
01:16:09,420 --> 01:16:12,750
You just put arbitrary
complex numbers

1058
01:16:12,750 --> 01:16:17,980
and require that the matrix
be equal to its matrix complex

1059
01:16:17,980 --> 01:16:19,420
conjugate and transpose.

1060
01:16:19,420 --> 01:16:21,245
So this is the most general one.

1061
01:16:21,245 --> 01:16:25,210
Do I obtain this
matrix from this one's?

1062
01:16:25,210 --> 01:16:34,440
Yes I just have to put 1 times
c plus a times sigma 1 plus b

1063
01:16:34,440 --> 01:16:38,990
times sigma 2 plus
d times sigma 3.

1064
01:16:38,990 --> 01:16:44,840
So any Hermitian
matrix can be obtained

1065
01:16:44,840 --> 01:16:47,150
as the span of this list.

1066
01:16:50,160 --> 01:16:53,630
Is this list
linearly independent?

1067
01:16:53,630 --> 01:16:58,350
So I have to go here
and set this equal to 0

1068
01:16:58,350 --> 01:17:03,890
and see if it sets to 0
all these coefficients.

1069
01:17:03,890 --> 01:17:08,510
Well, it's the same thing as
setting to 0 all this matrix.

1070
01:17:08,510 --> 01:17:15,160
Well, if c plus d and c minus
d are 0, then c and d are 0.

1071
01:17:15,160 --> 01:17:20,030
If this is 0, it must be a 0
and b 0, so all of them are 0.

1072
01:17:20,030 --> 01:17:22,970
So yes, it's
linearly independent.

1073
01:17:22,970 --> 01:17:24,440
It spans.

1074
01:17:24,440 --> 01:17:27,890
Therefore, you've proven
completely rigorously

1075
01:17:27,890 --> 01:17:32,525
that this vector
space is dimension 4.

1076
01:17:41,940 --> 01:17:45,270
This vector space--
I will actually

1077
01:17:45,270 --> 01:17:49,890
leave it as an exercise for
you to show that this vector

1078
01:17:49,890 --> 01:17:51,214
space is infinite dimensional.

1079
01:17:51,214 --> 01:17:53,130
You say, of course, it's
infinite dimensional.

1080
01:17:53,130 --> 01:17:55,530
It has infinite sequences.

1081
01:17:55,530 --> 01:17:58,000
Well, you have to
show that if you

1082
01:17:58,000 --> 01:18:01,740
have a finite list of
those infinite sequences,

1083
01:18:01,740 --> 01:18:07,240
like 300 sequences,
they span that.

1084
01:18:07,240 --> 01:18:08,650
They cannot span that.

1085
01:18:08,650 --> 01:18:12,620
So it takes a little work.

1086
01:18:12,620 --> 01:18:14,235
It's interesting
to think about it.

1087
01:18:14,235 --> 01:18:18,510
I think you will enjoy trying
to think about this stuff.

1088
01:18:18,510 --> 01:18:24,400
So that's our discussion
of dimensionality.

1089
01:18:24,400 --> 01:18:29,620
So this one is a little
harder to make sure

1090
01:18:29,620 --> 01:18:31,100
it's infinite dimensional.

1091
01:18:31,100 --> 01:18:34,310
And this one is, yet, a
bit harder than that one

1092
01:18:34,310 --> 01:18:36,210
but it can also be done.

1093
01:18:36,210 --> 01:18:37,535
This is infinite dimensional.

1094
01:18:40,356 --> 01:18:41,730
And this is infinite
dimensional.

1095
01:18:44,600 --> 01:18:49,300
In the last two minute, I want
to tell you a little bit-- one

1096
01:18:49,300 --> 01:18:53,130
definition and let
you go with that,

1097
01:18:53,130 --> 01:18:56,413
is the definition of
a linear operator.

1098
01:19:01,280 --> 01:19:03,130
So here is one thing.

1099
01:19:03,130 --> 01:19:09,270
So you can be more general,
and we won't be that general.

1100
01:19:09,270 --> 01:19:13,350
But when you talk
about linear maps,

1101
01:19:13,350 --> 01:19:20,980
you have one vector space and
another vector space, v and w.

1102
01:19:20,980 --> 01:19:26,930
This is a vector space and
this is a vector space.

1103
01:19:26,930 --> 01:19:31,640
And in general, a map from
here is sometimes called,

1104
01:19:31,640 --> 01:19:34,610
if it satisfies the
property, a linear map.

1105
01:19:37,120 --> 01:19:39,970
And the key thing is
that in all generality,

1106
01:19:39,970 --> 01:19:43,695
these two vector spaces may
not have the same dimension.

1107
01:19:43,695 --> 01:19:46,901
It might be one vector space and
another very different vector

1108
01:19:46,901 --> 01:19:47,400
space.

1109
01:19:47,400 --> 01:19:50,230
You go from one to the other.

1110
01:19:50,230 --> 01:19:54,530
Now, when you have
a vector space v

1111
01:19:54,530 --> 01:19:57,770
and you map to the
same vector space,

1112
01:19:57,770 --> 01:20:00,060
this is also a
linear map, but this

1113
01:20:00,060 --> 01:20:04,860
is called an operator
or a linear operator.

1114
01:20:07,600 --> 01:20:10,980
And what is a linear
operator therefore?

1115
01:20:10,980 --> 01:20:20,310
A linear operator is
a function T. Let's

1116
01:20:20,310 --> 01:20:26,480
call the linear operator T.
It takes v to v. In which way?

1117
01:20:26,480 --> 01:20:35,110
Well, T acting u plus v,
on the sum of vectors,

1118
01:20:35,110 --> 01:20:44,790
is Tu plus T v. And T
acting on a times a vector

1119
01:20:44,790 --> 01:20:49,380
is a times T of the vector.

1120
01:20:49,380 --> 01:20:53,160
These two things make
it into something

1121
01:20:53,160 --> 01:20:55,710
we call a linear operator.

1122
01:20:55,710 --> 01:21:00,400
It acts on the sum
of vectors linearly

1123
01:21:00,400 --> 01:21:02,900
and on a number times a vector.

1124
01:21:02,900 --> 01:21:06,236
The number goes out and
you act on the vector.

1125
01:21:06,236 --> 01:21:11,300
So all you need to know for
what a linear operator is,

1126
01:21:11,300 --> 01:21:14,890
is how it acts on basis vectors.

1127
01:21:14,890 --> 01:21:17,620
Because any vector
on the vector space

1128
01:21:17,620 --> 01:21:19,800
is a superposition
of basis vectors.

1129
01:21:19,800 --> 01:21:23,410
So if you tell me how it
acts on the basis vectors,

1130
01:21:23,410 --> 01:21:24,960
you know everything.

1131
01:21:24,960 --> 01:21:29,000
So we will figure out how
the matrix representation

1132
01:21:29,000 --> 01:21:34,530
of the operators arises from how
it acts on the basis vectors.

1133
01:21:34,530 --> 01:21:36,940
And you don't need
an inner product.

1134
01:21:36,940 --> 01:21:39,460
The reason people think
of this is they say,

1135
01:21:39,460 --> 01:21:44,970
oh, the T i j
matrix element of T

1136
01:21:44,970 --> 01:21:49,310
is the inner product of the
operator between i and j.

1137
01:21:49,310 --> 01:21:51,960
And this is true.

1138
01:21:51,960 --> 01:21:54,340
But for that you
need [? brass ?]

1139
01:21:54,340 --> 01:21:56,690
and inner product,
all these things.

1140
01:21:56,690 --> 01:21:58,460
And they're not necessary.

1141
01:21:58,460 --> 01:22:00,620
We'll define this without that.

1142
01:22:00,620 --> 01:22:02,100
We don't need it.

1143
01:22:02,100 --> 01:22:06,640
So see you next time,
and we'll continue that.

1144
01:22:06,640 --> 01:22:09,940
[APPLAUSE]

1145
01:22:09,940 --> 01:22:11,790
Thank you.