1
00:00:00,170 --> 00:00:01,070
- [Instructor] In a previous video,

2
00:00:01,070 --> 00:00:02,240
we saw that if you have a system

3
00:00:02,240 --> 00:00:04,900
of three equations with
three unknowns like this,

4
00:00:04,900 --> 00:00:08,360
you can represent it as
a matrix vector equation,

5
00:00:08,360 --> 00:00:10,520
where this matrix right over here

6
00:00:10,520 --> 00:00:13,740
is a three-by-three matrix.

7
00:00:13,740 --> 00:00:16,180
That is essentially a coefficient matrix.

8
00:00:16,180 --> 00:00:18,300
It has all of the coefficients of the Xs,

9
00:00:18,300 --> 00:00:21,350
the Ys, and the Zs as its various columns.

10
00:00:21,350 --> 00:00:24,660
and then you're going to
multiply that times this vector,

11
00:00:24,660 --> 00:00:27,070
which is really the vector
of the unknown variables,

12
00:00:27,070 --> 00:00:29,570
and this is a three-by-one vector.

13
00:00:29,570 --> 00:00:30,830
And then you would result

14
00:00:30,830 --> 00:00:35,450
in this other three-by-one
vector, which is a vector

15
00:00:35,450 --> 00:00:40,050
that contains these constant
terms right over here.

16
00:00:40,050 --> 00:00:42,160
What we're gonna do in
this video is recognize

17
00:00:42,160 --> 00:00:44,570
that you can generalize this phenomenon.

18
00:00:44,570 --> 00:00:46,340
It's not just true with a system

19
00:00:46,340 --> 00:00:48,580
of three equations with three unknowns.

20
00:00:48,580 --> 00:00:52,490
It actually generalizes to
N equations with N unknowns.

21
00:00:52,490 --> 00:00:55,600
But just to appreciate that
that is indeed the case,

22
00:00:55,600 --> 00:00:59,610
let us look at a system of two
equations with two unknowns.

23
00:00:59,610 --> 00:01:04,020
So let's say you had 2x
plus y is equal to nine,

24
00:01:04,020 --> 00:01:09,020
and we had 3x minus y is equal to five.

25
00:01:09,230 --> 00:01:10,630
I encourage you, pause this video

26
00:01:10,630 --> 00:01:12,610
and think about how that
would be represented

27
00:01:12,610 --> 00:01:15,253
as a matrix vector equation.

28
00:01:16,520 --> 00:01:19,520
All right, now let's
work on this together.

29
00:01:19,520 --> 00:01:23,710
So this is a system of two
equations with two unknowns.

30
00:01:23,710 --> 00:01:27,010
So the matrix that represents
the coefficients is going

31
00:01:27,010 --> 00:01:29,390
to be a two-by-two matrix

32
00:01:29,390 --> 00:01:31,680
and then that's going to be multiplied

33
00:01:31,680 --> 00:01:35,200
by a vector that represents
the unknown variables.

34
00:01:35,200 --> 00:01:37,090
We have two unknown variables over here.

35
00:01:37,090 --> 00:01:40,120
So this is going to be
a two-by-one vector,

36
00:01:40,120 --> 00:01:42,910
and then that's going
to be equal to a vector

37
00:01:42,910 --> 00:01:45,520
that represents the constants
on the right-hand side,

38
00:01:45,520 --> 00:01:46,810
and obviously we have two of those.

39
00:01:46,810 --> 00:01:49,430
So that's going to be a
two-by-one vector as well.

40
00:01:49,430 --> 00:01:51,870
And then we can do exactly what we did

41
00:01:51,870 --> 00:01:54,410
in that previous example
in a previous video.

42
00:01:54,410 --> 00:01:58,850
The coefficients on the
X terms, two and three,

43
00:01:58,850 --> 00:02:02,570
and then we have the
coefficients on the Y terms.

44
00:02:02,570 --> 00:02:04,530
This would be a positive one

45
00:02:04,530 --> 00:02:06,930
and then this would be a negative one.

46
00:02:06,930 --> 00:02:09,570
And then we multiply it times the vector

47
00:02:09,570 --> 00:02:12,490
of the variables, X, Y,

48
00:02:12,490 --> 00:02:14,700
and then last but not least,

49
00:02:14,700 --> 00:02:19,200
you have this nine and this
five over here, nine and five.

50
00:02:19,200 --> 00:02:22,040
And I encourage you and multiply this out.

51
00:02:22,040 --> 00:02:25,390
Multiply this matrix times this vector.

52
00:02:25,390 --> 00:02:28,560
And when you do that and you
still set up this equality,

53
00:02:28,560 --> 00:02:30,620
you're going to see that
it essentially turns

54
00:02:30,620 --> 00:02:32,770
into this exact same system

55
00:02:32,770 --> 00:02:36,010
of two equations and two unknowns.

56
00:02:36,010 --> 00:02:37,650
Now, what's interesting about this is

57
00:02:37,650 --> 00:02:40,730
that we see a generalizable form.

58
00:02:40,730 --> 00:02:43,860
In general, you can represent a system

59
00:02:43,860 --> 00:02:47,890
of N equations and N unknowns in the form.

60
00:02:47,890 --> 00:02:51,740
Sum N-by-N matrix A,

61
00:02:51,740 --> 00:02:53,400
N by N,

62
00:02:53,400 --> 00:02:58,340
times sum N-by-one vector X.

63
00:02:58,340 --> 00:02:59,523
This isn't just the variable X.

64
00:02:59,523 --> 00:03:03,950
This is a vector X that
has N dimensions to it.

65
00:03:03,950 --> 00:03:08,610
So times sum N-by-one vector X is going

66
00:03:08,610 --> 00:03:13,530
to be equal to sum N-by-one vector B.

67
00:03:14,480 --> 00:03:17,360
These are the letters that
people use by convention.

68
00:03:17,360 --> 00:03:19,030
This is going to be N by one.

69
00:03:19,030 --> 00:03:21,890
And so you can see in
these different scenarios.

70
00:03:21,890 --> 00:03:24,850
In that first one, this is
a three-by-three matrix.

71
00:03:24,850 --> 00:03:26,870
We could call that A,

72
00:03:26,870 --> 00:03:29,580
and then we could call this the vector X,

73
00:03:29,580 --> 00:03:31,760
and then we could call this the vector B.

74
00:03:31,760 --> 00:03:32,850
Now in that second scenario,

75
00:03:32,850 --> 00:03:34,800
we could call this the matrix A,

76
00:03:34,800 --> 00:03:36,850
we could call this the vector X,

77
00:03:36,850 --> 00:03:39,320
and then we could call this the vector B,

78
00:03:39,320 --> 00:03:42,820
but we can generalize
that to N dimensions.

79
00:03:42,820 --> 00:03:44,760
And as I talked about
in the previous video,

80
00:03:44,760 --> 00:03:47,880
what's interesting about this
is you could think about,

81
00:03:47,880 --> 00:03:49,890
for example, in this
system of two equations

82
00:03:49,890 --> 00:03:52,300
with two unknowns, as all
right, I have a line here,

83
00:03:52,300 --> 00:03:53,490
I have a line here,

84
00:03:53,490 --> 00:03:57,640
and X and Y represent the
intersection of those lines.

85
00:03:57,640 --> 00:03:58,850
But when you represent it this way,

86
00:03:58,850 --> 00:04:00,580
you could also imagine it as saying,

87
00:04:00,580 --> 00:04:04,470
okay, I have some unknown
vector in the coordinate plane

88
00:04:04,470 --> 00:04:07,460
and I'm transforming it using this matrix

89
00:04:07,460 --> 00:04:10,030
to get this vector nine five.

90
00:04:10,030 --> 00:04:12,410
And so I have to figure out what vector,

91
00:04:12,410 --> 00:04:15,510
when transformed in this
way, gets us to nine five,

92
00:04:15,510 --> 00:04:18,160
and we also thought about it
in the three-by-three case.

93
00:04:18,160 --> 00:04:20,950
What three-dimensional vector,
when transformed in this way,

94
00:04:20,950 --> 00:04:23,360
gets us to this vector right over here?

95
00:04:23,360 --> 00:04:24,910
And so that hints,

96
00:04:24,910 --> 00:04:27,410
that foreshadows where
we might be able to go.

97
00:04:27,410 --> 00:04:30,530
If we can unwind this
transformation somehow,

98
00:04:30,530 --> 00:04:33,850
then we can figure out what
these unknown vectors are.

99
00:04:33,850 --> 00:04:37,270
And if we can do it in two
dimensions or three dimensions,

100
00:04:37,270 --> 00:04:40,010
why not be able to do it in N dimensions?

101
00:04:40,010 --> 00:04:41,980
Which you'll see is actually very useful

102
00:04:41,980 --> 00:04:43,650
if you ever become a data scientist,

103
00:04:43,650 --> 00:04:45,220
or you go into computer science,

104
00:04:45,220 --> 00:04:47,670
or if you go into computer
graphics of some kind.