1
00:00:00,490 --> 00:00:03,320
I'm going to do one more video
where we compare old and new

2
00:00:03,320 --> 00:00:05,320
definitions of a projection.

3
00:00:05,320 --> 00:00:10,560
Our old definition of a
projection onto some line, l,

4
00:00:10,560 --> 00:00:18,380
of the vector, x, is the vector
in l, or that's a

5
00:00:18,380 --> 00:00:26,500
member of l, such that x minus
that vector, minus the

6
00:00:26,500 --> 00:00:38,540
projection onto l of x,
is orthogonal to l.

7
00:00:38,540 --> 00:00:41,990
So the visualization is, if
you have your line l like

8
00:00:41,990 --> 00:00:46,360
this, that is your line
l right there.

9
00:00:46,360 --> 00:00:49,950
And then you have some other
vector x that we're take the

10
00:00:49,950 --> 00:00:52,010
projection of it on to l.

11
00:00:52,010 --> 00:00:53,450
So that's x.

12
00:00:53,450 --> 00:00:56,770
The projection of x onto l,
this thing right here, is

13
00:00:56,770 --> 00:01:00,490
going to be some vector in l.

14
00:01:00,490 --> 00:01:03,060
Such that when I take the
difference between x and that

15
00:01:03,060 --> 00:01:05,880
vector, it's going to
be orthogonal to l.

16
00:01:05,880 --> 00:01:07,940
So it's going to be
some vector in l.

17
00:01:07,940 --> 00:01:10,190
This was our old definition when
we took the projection

18
00:01:10,190 --> 00:01:10,990
onto a line.

19
00:01:10,990 --> 00:01:12,460
Some vector in l.

20
00:01:12,460 --> 00:01:13,460
Maybe it's there.

21
00:01:13,460 --> 00:01:16,090
And if I take the difference
between that and that, this

22
00:01:16,090 --> 00:01:18,090
difference vector's going
to be orthogonal to

23
00:01:18,090 --> 00:01:19,340
everything in l.

24
00:01:23,080 --> 00:01:26,110
Just like that.

25
00:01:26,110 --> 00:01:28,700
So, this right here would be
it's difference vector.

26
00:01:28,700 --> 00:01:33,650
That would be x minus the
projection of x onto l.

27
00:01:33,650 --> 00:01:36,200
And then, of course, this
vector right here.

28
00:01:36,200 --> 00:01:39,950
This is the one we
were defining it.

29
00:01:39,950 --> 00:01:44,350
That was the projection
onto l of x.

30
00:01:44,350 --> 00:01:46,270
Now, what's a different
way that we could

31
00:01:46,270 --> 00:01:48,640
have written this?

32
00:01:48,640 --> 00:01:51,230
We could have written this
exact same definition.

33
00:01:51,230 --> 00:01:56,200
We could have said it is the
vector in l such that-- so we

34
00:01:56,200 --> 00:01:58,460
could say, let me write
it here in purple.

35
00:01:58,460 --> 00:02:14,010
Is the vector v in l such that
v-- let me write it this way--

36
00:02:14,010 --> 00:02:18,215
such that x minus v, right?

37
00:02:18,215 --> 00:02:24,340
x minus the projection of l is
orthogonal is equal to w,

38
00:02:24,340 --> 00:02:36,660
which is orthogonal to
everything in l.

39
00:02:36,660 --> 00:02:38,730
Being orthogonal to l literally
means being

40
00:02:38,730 --> 00:02:41,050
orthogonal to every
vector in l.

41
00:02:41,050 --> 00:02:43,180
So I just rewrote it a little
bit different, instead of just

42
00:02:43,180 --> 00:02:45,800
leaving it as a projection
of x onto l.

43
00:02:45,800 --> 00:02:51,780
I said hey, that's some vector,
v, in l such that x

44
00:02:51,780 --> 00:02:58,310
minus v is equal to some other
vector, w, which is orthogonal

45
00:02:58,310 --> 00:02:59,660
to everything in l.

46
00:02:59,660 --> 00:03:05,090
Or we can rewrite that statement
right there as x is

47
00:03:05,090 --> 00:03:08,250
equal to v plus w.

48
00:03:08,250 --> 00:03:12,670
So we can just say that the
projection of x onto l is the

49
00:03:12,670 --> 00:03:19,470
unique vector v in l, such that
x is equal to v plus w,

50
00:03:19,470 --> 00:03:23,590
where w is a unique vector-- I
mean it is going to be unique

51
00:03:23,590 --> 00:03:27,460
vector-- in the orthogonal
complement of l.

52
00:03:27,460 --> 00:03:28,080
Right?

53
00:03:28,080 --> 00:03:31,070
This is got to be orthogonal
to everything in l.

54
00:03:31,070 --> 00:03:33,410
So that's going to be a member
of the orthogonal

55
00:03:33,410 --> 00:03:38,070
complement of l.

56
00:03:38,070 --> 00:03:40,910
So this definition is actually
completely consistent with our

57
00:03:40,910 --> 00:03:42,950
new subspace definition.

58
00:03:42,950 --> 00:03:45,820
And we could just extend
it to arbitrary

59
00:03:45,820 --> 00:03:47,410
subspaces, not just lines.

60
00:03:47,410 --> 00:03:51,390
Let me help you visualize
that.

61
00:03:51,390 --> 00:03:55,630
So let's say we're dealing
with R3 right here.

62
00:03:59,590 --> 00:04:02,890
And I've got some
subspace in R3.

63
00:04:02,890 --> 00:04:04,950
And let's say that subspace
happens to be a plane.

64
00:04:04,950 --> 00:04:08,140
I'm going to make it a plane
just so that it becomes clear

65
00:04:08,140 --> 00:04:10,290
that we don't have to take
projections just onto lines.

66
00:04:10,290 --> 00:04:13,920
So this is my subspace
v right there.

67
00:04:13,920 --> 00:04:16,140
Let me draw its orthogonal
compliment.

68
00:04:16,140 --> 00:04:18,500
Let's say its orthogonal
complement looks

69
00:04:18,500 --> 00:04:20,610
something like that.

70
00:04:20,610 --> 00:04:22,100
Let's say it's a line.

71
00:04:22,100 --> 00:04:24,300
And then it goes-- it intersects
right there.

72
00:04:24,300 --> 00:04:25,520
Then it goes back.

73
00:04:25,520 --> 00:04:28,360
And, of course, it would have to
intersect at the 0 vector.

74
00:04:28,360 --> 00:04:33,170
That's the only place where a
subspace and its orthogonal

75
00:04:33,170 --> 00:04:34,160
complement overlap.

76
00:04:34,160 --> 00:04:36,710
And then it goes behind
and you see it again.

77
00:04:36,710 --> 00:04:38,860
Obviously you wouldn't be able
to you again because this

78
00:04:38,860 --> 00:04:40,540
plane would extend in
every direction.

79
00:04:40,540 --> 00:04:41,770
But you get the idea.

80
00:04:41,770 --> 00:04:45,330
So this right here
is the orthogonal

81
00:04:45,330 --> 00:04:48,550
complement of v, that line.

82
00:04:48,550 --> 00:04:53,510
Now, let's have some other
arbitrary vector in R3 here.

83
00:04:53,510 --> 00:04:56,850
So let's say I have some vector
that looks like that.

84
00:04:56,850 --> 00:04:59,180
Let's say that that is x.

85
00:04:59,180 --> 00:05:08,670
Now our new definition for the
projection of x onto v is

86
00:05:08,670 --> 00:05:16,540
equal to the unique vector v.

87
00:05:16,540 --> 00:05:17,390
This is a vector v.

88
00:05:17,390 --> 00:05:18,400
That's a subspace v.

89
00:05:18,400 --> 00:05:27,240
The unique vector v, that is a
member of v, such that x is

90
00:05:27,240 --> 00:05:43,120
equal to v plus w, where w
is a unique member of the

91
00:05:43,120 --> 00:05:45,610
orthogonal complement of v.

92
00:05:45,610 --> 00:05:47,120
This is our new definition.

93
00:05:47,120 --> 00:05:51,250
So, if we say x is equal to
some member of v and some

94
00:05:51,250 --> 00:05:53,490
member of its orthogonal
complement-- we can visually

95
00:05:53,490 --> 00:05:54,710
understand that here.

96
00:05:54,710 --> 00:05:59,280
We could say, OK it's going to
be equal to, on v, it'll be

97
00:05:59,280 --> 00:06:01,470
equal to that vector
to right there.

98
00:06:01,470 --> 00:06:04,410
And then on v's orthogonal
complement,

99
00:06:04,410 --> 00:06:05,670
you add that to it.

100
00:06:05,670 --> 00:06:07,470
So, if you were to shift it
over, you would get that

101
00:06:07,470 --> 00:06:09,010
vector, just like that.

102
00:06:09,010 --> 00:06:11,310
This right here is v.

103
00:06:11,310 --> 00:06:12,750
That right there is v.

104
00:06:12,750 --> 00:06:15,950
And then this is vector that
goes up like this, out of the

105
00:06:15,950 --> 00:06:18,340
plane, orthogonal to
the plane, is w.

106
00:06:18,340 --> 00:06:21,940
You could see if you take v plus
w, you're going to get x.

107
00:06:21,940 --> 00:06:29,400
And you could see that v is
the projection onto the

108
00:06:29,400 --> 00:06:33,610
subspace capital v-- so this
is a vector, v-- is the

109
00:06:33,610 --> 00:06:38,600
projection onto the subspace
capital V of the vector x.

110
00:06:38,600 --> 00:06:40,710
So the analogy to a shadow
still holds.

111
00:06:40,710 --> 00:06:43,640
If you imagine kind of a light
source coming straight down

112
00:06:43,640 --> 00:06:47,060
onto our subspace, kind of
orthogonal to our subspace,

113
00:06:47,060 --> 00:06:50,570
the projection onto our subspace
is kind of the shadow

114
00:06:50,570 --> 00:06:51,320
of our vector x.

115
00:06:51,320 --> 00:06:53,880
Hopefully that help you
visualize it a little better.

116
00:06:53,880 --> 00:06:57,430
But what we're doing here is
we're going to generalize it.

117
00:06:57,430 --> 00:06:58,960
Earlier in this video
I showed you a line.

118
00:06:58,960 --> 00:07:00,310
This is a plane.

119
00:07:00,310 --> 00:07:02,000
But we can generalize
it to any subspace.

120
00:07:02,000 --> 00:07:02,920
This is in R3.

121
00:07:02,920 --> 00:07:06,230
We can generalize it
to Rn, to R100.

122
00:07:06,230 --> 00:07:08,630
And that's really the power
of what we're doing here.

123
00:07:08,630 --> 00:07:11,360
It's easy to visualize it here,
but it's not so easy to

124
00:07:11,360 --> 00:07:13,530
visualize it once you get
to higher dimensions.

125
00:07:13,530 --> 00:07:14,900
And actually, one other thing.

126
00:07:14,900 --> 00:07:17,370
Let me show that this new
definition is pretty much

127
00:07:17,370 --> 00:07:20,130
almost identical to exactly
what we did with lines.

128
00:07:20,130 --> 00:07:24,950
This is identical to saying that
the projection onto the

129
00:07:24,950 --> 00:07:45,820
subspace x is equal to some
unique vector in V such that x

130
00:07:45,820 --> 00:07:58,290
minus the projection onto v of
x is orthogonal to every

131
00:07:58,290 --> 00:08:01,900
member of V.

132
00:08:01,900 --> 00:08:05,230
Because this statement, right
here, is saying any vector

133
00:08:05,230 --> 00:08:08,250
that's orthogonal to any member
of v says that it's a

134
00:08:08,250 --> 00:08:10,610
member of the orthogonal
complement of v.

135
00:08:10,610 --> 00:08:13,830
So that statement could be
written as x minus the

136
00:08:13,830 --> 00:08:20,450
projection onto v of x is a
member of v's orthogonal

137
00:08:20,450 --> 00:08:21,030
complement.

138
00:08:21,030 --> 00:08:23,310
Or we could call some w.

139
00:08:23,310 --> 00:08:27,210
So if you call this your v,
and if you call this whole

140
00:08:27,210 --> 00:08:31,640
thing your w, you get this exact
definition right there.

141
00:08:31,640 --> 00:08:35,820
You would have w is equal
to x minus v.

142
00:08:35,820 --> 00:08:39,840
And then if you add v to both
sides, you get w plus v is

143
00:08:39,840 --> 00:08:40,950
equal to x.

144
00:08:40,950 --> 00:08:44,250
We defined v to be, the
orthogonal-- the

145
00:08:44,250 --> 00:08:46,250
projection of x onto v.

146
00:08:46,250 --> 00:08:51,050
w is a member of our orthogonal
complement.

147
00:08:51,050 --> 00:08:53,990
And I don't want you
to get confused.

148
00:08:53,990 --> 00:08:57,560
The vector v is the orthogonal
projection of our vector x

149
00:08:57,560 --> 00:09:00,080
onto the subspace capital V.

150
00:09:00,080 --> 00:09:02,170
I probably should use different
letters instead of

151
00:09:02,170 --> 00:09:03,740
using a lowercase and
a uppercase v.

152
00:09:03,740 --> 00:09:05,540
It makes the language
a little difficult.

153
00:09:05,540 --> 00:09:07,890
But I just wanted to give you
another video to give you a

154
00:09:07,890 --> 00:09:10,040
visualization of projections
onto

155
00:09:10,040 --> 00:09:11,650
subspaces other than lines.

156
00:09:11,650 --> 00:09:14,570
And to show you that our old
definition, with just a

157
00:09:14,570 --> 00:09:17,070
projection onto a line which was
a linear transformation,

158
00:09:17,070 --> 00:09:19,400
is essentially equivalent
to this new definition.

159
00:09:19,400 --> 00:09:23,400
On the next video, I'll show
you that this, for any

160
00:09:23,400 --> 00:09:26,740
subspace is, indeed, a linear
transformation.