1
00:00:13,500 --> 00:00:15,720
GILBERT STRANG:
Well, I'm Gil Strang.

2
00:00:15,720 --> 00:00:20,650
And I'm very happy if you
know the linear algebra

3
00:00:20,650 --> 00:00:24,620
videos on OpenCourseWare
or on YouTube.

4
00:00:24,620 --> 00:00:28,750
That's for the
math course 18.06.

5
00:00:28,750 --> 00:00:32,409
And I'm even happier
if you like them.

6
00:00:32,409 --> 00:00:38,590
And I'm here today to update
them for several reasons.

7
00:00:38,590 --> 00:00:45,320
Well, a lots happened in
linear algebra in these years.

8
00:00:45,320 --> 00:00:48,280
Fortunately, the
whole subject has just

9
00:00:48,280 --> 00:00:51,760
become more and more essential,
more and more important, more

10
00:00:51,760 --> 00:00:53,380
and more beautiful.

11
00:00:53,380 --> 00:00:59,650
And so I wanted to say something
about those later steps.

12
00:00:59,650 --> 00:01:05,710
And also, when I teach it now,
I have a new starting point.

13
00:01:05,710 --> 00:01:06,980
And I'll show you that.

14
00:01:06,980 --> 00:01:10,110
So I'll go a little slowly
on that starting point.

15
00:01:10,110 --> 00:01:12,460
The slides tell you
the whole course.

16
00:01:12,460 --> 00:01:18,340
And that's crazy to have a full
course within a short video.

17
00:01:18,340 --> 00:01:23,650
But especially the
first part is new.

18
00:01:23,650 --> 00:01:27,580
And I'm even writing
a textbook called

19
00:01:27,580 --> 00:01:31,310
Linear Algebra for Everyone
that will start this way.

20
00:01:31,310 --> 00:01:39,970
I hope that the new start brings
in real linear algebra ideas

21
00:01:39,970 --> 00:01:41,030
right away.

22
00:01:41,030 --> 00:01:42,700
And let me show you
where those are.

23
00:01:42,700 --> 00:01:47,800
So this is an outline
of the whole video.

24
00:01:47,800 --> 00:01:51,640
And the first
line, which I think

25
00:01:51,640 --> 00:01:55,930
about in my mind as
matrix A, is a product

26
00:01:55,930 --> 00:02:00,730
of a C matrix, a column
matrix, and a row matrix R.

27
00:02:00,730 --> 00:02:02,600
And you'll see what those are.

28
00:02:02,600 --> 00:02:06,380
So that's the new idea.

29
00:02:06,380 --> 00:02:08,080
That will come first.

30
00:02:08,080 --> 00:02:15,430
Then these are five
famous essential shorthand

31
00:02:15,430 --> 00:02:19,720
descriptions of the key
chapters of linear algebra,

32
00:02:19,720 --> 00:02:20,710
the key chapters.

33
00:02:20,710 --> 00:02:25,720
So they represent, for example,
LU is those letters are famous.

34
00:02:25,720 --> 00:02:28,930
And computer commands
would be exactly

35
00:02:28,930 --> 00:02:34,430
those letters LU for elimination
for solving equations,

36
00:02:34,430 --> 00:02:36,710
the first job of linear algebra.

37
00:02:36,710 --> 00:02:39,400
And then QR.

38
00:02:39,400 --> 00:02:43,660
So Q is a very interesting
important type of matrix.

39
00:02:43,660 --> 00:02:46,150
That's standing for
an orthogonal matrix.

40
00:02:46,150 --> 00:02:50,650
There is the word
orthogonal or perpendicular.

41
00:02:50,650 --> 00:02:54,340
So those are the best
matrices to compute with.

42
00:02:54,340 --> 00:02:58,250
And that QR gets us there.

43
00:02:58,250 --> 00:03:00,640
And S is for a symmetric matrix.

44
00:03:00,640 --> 00:03:02,100
And it will involve--

45
00:03:02,100 --> 00:03:04,570
oh, well, I should
say, the first half,

46
00:03:04,570 --> 00:03:11,650
ending with there, with QR,
is about solving equations.

47
00:03:11,650 --> 00:03:16,270
The second half, these
three are about eigenvalues

48
00:03:16,270 --> 00:03:18,460
and eigenvectors
and singular values,

49
00:03:18,460 --> 00:03:23,050
a different way to approach the
whole subject and a very, very

50
00:03:23,050 --> 00:03:24,760
important way.

51
00:03:24,760 --> 00:03:30,310
Among my goals is to help
courses around the world

52
00:03:30,310 --> 00:03:34,720
get singular values
included because you really

53
00:03:34,720 --> 00:03:36,460
don't want to miss them.

54
00:03:36,460 --> 00:03:39,340
That's the high
point of the theory.

55
00:03:39,340 --> 00:03:42,740
And it's expressed
like all the others

56
00:03:42,740 --> 00:03:48,580
as breaking a matrix into
two or three pieces, two

57
00:03:48,580 --> 00:03:49,930
or three parts.

58
00:03:49,930 --> 00:03:54,130
So that's my plan
for this video.

59
00:03:54,130 --> 00:03:56,140
And I hope it's helpful.

60
00:03:56,140 --> 00:04:00,320
Again, it's a whole
course in a short time.

61
00:04:00,320 --> 00:04:08,240
And please go to the real
18.06 videos for the details.

62
00:04:08,240 --> 00:04:09,490
Thanks