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Remember last time -- -- we
learned about the cross product
8
00:00:34 --> 00:00:42
of vectors in space.
Remember the definition of
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00:00:42 --> 00:00:48
cross product is in terms of
this determinant det| i hat,
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00:00:48 --> 00:00:53
j hat, k hat,
and then the components of A,
11
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a1, a2, a3,
and then the components of B|
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00:00:57 --> 00:01:02
This is not an actual
determinant because these are
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not numbers.
But it's a symbolic notation,
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00:01:05 --> 00:01:08
to remember what the actual
formula is.
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00:01:08 --> 00:01:12
The actual formula is obtained
by expanding the determinant.
16
00:01:12 --> 00:01:19
So, we actually get the
determinant of a2,
17
00:01:19 --> 00:01:27
a3, b2, b3 times i hat,
minus the determinant of a1,
18
00:01:27 --> 00:01:35
a3, b1, b3 times j hat plus the
determinant of a1,
19
00:01:35 --> 00:01:42
a2, b1, b2, times k hat.
And we also saw a more
20
00:01:42 --> 00:01:47
geometric definition of the
cross product.
21
00:01:47 --> 00:01:56
We have learned that the length
of the cross product is equal to
22
00:01:56 --> 00:02:04
the area of the parallelogram
with sides A and B.
23
00:02:04 --> 00:02:17
24
00:02:17 --> 00:02:26
We have also learned that the
direction of the cross product
25
00:02:26 --> 00:02:37
is given by taking the direction
that's perpendicular to A and B.
26
00:02:37 --> 00:02:42
If I draw A and B in a plane
(they determine a plane),
27
00:02:42 --> 00:02:48
then the cross product should
go in the direction that's
28
00:02:48 --> 00:02:53
perpendicular to that plane.
Now, there's two different
29
00:02:53 --> 00:02:56
possible directions that are
perpendicular to a plane.
30
00:02:56 --> 00:03:04
And, to decide which one it is,
we use the right-hand rule,
31
00:03:04 --> 00:03:07
which says if you extend your
right hand in the direction of
32
00:03:07 --> 00:03:10
the vector A,
then curve your fingers in the
33
00:03:10 --> 00:03:14
direction of B,
then your thumb will go in the
34
00:03:14 --> 00:03:20
direction of the cross product.
One thing I didn't quite get to
35
00:03:20 --> 00:03:26
say last time is that there are
some funny manipulation rules.
36
00:03:26 --> 00:03:29
What are we allowed to do or
not do with cross products?
37
00:03:29 --> 00:03:35
So, let me tell you right away
the most surprising one if
38
00:03:35 --> 00:03:41
you've never seen it before:
A cross B and B cross A are not
39
00:03:41 --> 00:03:45
the same thing.
Why are they not the same thing?
40
00:03:45 --> 00:03:49
Well, one way to see it is to
think geometrically.
41
00:03:49 --> 00:03:52
The parallelogram still has the
same area, and it's still in the
42
00:03:52 --> 00:03:54
same plane.
So, the cross product is still
43
00:03:54 --> 00:03:58
perpendicular to the same plane.
But, what happens is that,
44
00:03:58 --> 00:04:01
if you try to apply the
right-hand rule but exchange the
45
00:04:01 --> 00:04:04
roles of A and B,
then you will either injure
46
00:04:04 --> 00:04:06
yourself,
or your thumb will end up
47
00:04:06 --> 00:04:08
pointing in the opposite
direction.
48
00:04:08 --> 00:04:12
So, in fact,
B cross A and A cross B are
49
00:04:12 --> 00:04:17
opposite of each other.
And you can check that in the
50
00:04:17 --> 00:04:19
formula because,
for example,
51
00:04:19 --> 00:04:22
the i component is a2 b3 minus
a3 b2.
52
00:04:22 --> 00:04:27
If you swap the roles of A and
B, you will also have to change
53
00:04:27 --> 00:04:30
the signs.
That's a slightly surprising
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00:04:30 --> 00:04:33
thing, but you will see one
easily adjusts to it.
55
00:04:33 --> 00:04:36
It just means one must resist
the temptation to write AxB
56
00:04:36 --> 00:04:40
equals BxA.
Whenever you do that,
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00:04:40 --> 00:04:45
put a minus sign.
Now, in particular,
58
00:04:45 --> 00:04:53
what happens if I do A cross A?
Well, I will get zero.
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00:04:53 --> 00:04:54
And, there's many ways to see
that.
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00:04:54 --> 00:04:58
One is to use the formula.
Also, you can see that the
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00:04:58 --> 00:05:02
parallelogram formed by A and A
is completely flat,
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00:05:02 --> 00:05:06
and it has area zero.
So, we get the zero vector.
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00:05:06 --> 00:05:17
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00:05:17 --> 00:05:20
Hopefully you got practice with
cross products,
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00:05:20 --> 00:05:23
and computing them,
in recitation yesterday.
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00:05:23 --> 00:05:29
Let me just point out one
important application of cross
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product that maybe you haven't
seen yet.
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Let's say that I'm given three
points in space,
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00:05:36 --> 00:05:39
and I want to find the equation
of the plane that contains them.
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00:05:39 --> 00:05:45
So, say I have P1,
P2, P3, three points in space.
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00:05:45 --> 00:05:48
They determine a plane,
at least if they are not
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00:05:48 --> 00:05:51
aligned, and we would like to
find the equation of the plane
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00:05:51 --> 00:05:56
that they determine.
That means, let's say that we
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00:05:56 --> 00:06:01
have a point,
P, in space with coordinates x,
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00:06:01 --> 00:06:07
y, z.
Well, to find the equation of
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00:06:07 --> 00:06:14
the plane -- -- the plane
containing P1,
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00:06:14 --> 00:06:22
P2, and P3,
we need to find a condition on
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00:06:22 --> 00:06:26
the coordinates x,
y, z,
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00:06:26 --> 00:06:41
telling us whether P is in the
plane or not.
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00:06:41 --> 00:06:44
We have several ways of doing
that.
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00:06:44 --> 00:06:47
For example,
one thing we could do.
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00:06:47 --> 00:06:51
Let me just backtrack to
determinants that we saw last
83
00:06:51 --> 00:06:56
time.
One way to think about it is to
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00:06:56 --> 00:07:03
consider these vectors,
P1P2, P1P3, and P1P.
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00:07:03 --> 00:07:07
The question of whether they
are all in the same plane is the
86
00:07:07 --> 00:07:12
same as asking ourselves whether
the parallelepiped that they
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00:07:12 --> 00:07:15
form is actually completely
flattened.
88
00:07:15 --> 00:07:18
So, if I try to form a
parallelepiped with these three
89
00:07:18 --> 00:07:21
sides, and P is not in the
plane, then it will have some
90
00:07:21 --> 00:07:24
volume.
But, if P is in the plane,
91
00:07:24 --> 00:07:26
then it's actually completely
squished.
92
00:07:26 --> 00:07:31
So,one possible answer,
one possible way to think of
93
00:07:31 --> 00:07:37
the equation of a plane is that
the determinant of these vectors
94
00:07:37 --> 00:07:42
should be zero.
Take the determinant of (vector
95
00:07:42 --> 00:07:48
P1P,vector P1P2,vector P1P3)
equals 0 (if you do it in a
96
00:07:48 --> 00:07:53
different order it doesn't
really matter).
97
00:07:53 --> 00:07:58
One possible way to express the
condition that P is in the plane
98
00:07:58 --> 00:08:02
is to say that the determinant
of these three vectors has to be
99
00:08:02 --> 00:08:05
zero.
And, if I am given coordinates
100
00:08:05 --> 00:08:07
for these points -- I'm not
giving you numbers,
101
00:08:07 --> 00:08:10
but if I gave you numbers,
then you would be able to plug
102
00:08:10 --> 00:08:14
those numbers in.
So, you could compute these two
103
00:08:14 --> 00:08:16
vectors P1P2 and P1P3
explicitly.
104
00:08:16 --> 00:08:19
But, of course,
P1P would depend on x,
105
00:08:19 --> 00:08:21
y, and z.
So, when you compute the
106
00:08:21 --> 00:08:24
determinant, you get a formula
that involves x,
107
00:08:24 --> 00:08:26
y, and z.
And you'll find that this
108
00:08:26 --> 00:08:29
condition on x,
y, z is the equation of a
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00:08:29 --> 00:08:32
plane.
We're going to see more about
110
00:08:32 --> 00:08:36
that pretty soon.
Now, let me tell you a slightly
111
00:08:36 --> 00:08:40
faster way of doing it.
Actually, it's not much faster,
112
00:08:40 --> 00:08:44
It's pretty much the same
calculation, but it's maybe more
113
00:08:44 --> 00:08:50
enlightening.
Let me actually show you a nice
114
00:08:50 --> 00:08:56
color picture that I prepared
for this.
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00:08:56 --> 00:09:00
One thing that's on this
picture that I haven't drawn
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00:09:00 --> 00:09:02
before is the normal vector to
the plane.
117
00:09:02 --> 00:09:06
Why is that?
Well, let's say that we know
118
00:09:06 --> 00:09:09
how to find a vector that's
perpendicular to our plane.
119
00:09:09 --> 00:09:13
Then, what does it mean for the
point, P, to be in the plane?
120
00:09:13 --> 00:09:19
It means that the direction
from P1 to P has to be
121
00:09:19 --> 00:09:29
perpendicular to this vector N.
So here's another solution:
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00:09:29 --> 00:09:43
P is in the plane exactly when
the vector P1P is perpendicular
123
00:09:43 --> 00:09:48
to N,
where N is some vector that's
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00:09:48 --> 00:10:05
perpendicular to the plane.
N is called a normal vector.
125
00:10:05 --> 00:10:08
How do we rephrase this
condition?
126
00:10:08 --> 00:10:13
Well, we've learned how to
detect whether two vectors are
127
00:10:13 --> 00:10:18
perpendicular to each other
using dot product (that was the
128
00:10:18 --> 00:10:21
first lecture).
These two vectors are
129
00:10:21 --> 00:10:25
perpendicular exactly when their
dot product is zero.
130
00:10:25 --> 00:10:32
So, concretely,
if we have a point P1 given to
131
00:10:32 --> 00:10:34
us,
and say we have been able to
132
00:10:34 --> 00:10:37
compute the vector N,
then when we actually compute
133
00:10:37 --> 00:10:40
what happens,
here we will have the
134
00:10:40 --> 00:10:41
coordinates x,
y, z, of a point P,
135
00:10:41 --> 00:10:44
and we will get some condition
on x, y, z.
136
00:10:44 --> 00:10:47
That will be the equation of a
plane.
137
00:10:47 --> 00:10:50
Now, why are these things the
same?
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00:10:50 --> 00:10:54
Well, before I can tell you
that, I should tell you how to
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00:10:54 --> 00:10:57
find a normal vector.
Maybe you are already starting
140
00:10:57 --> 00:11:01
to see what the method should
be, because we know how to find
141
00:11:01 --> 00:11:04
a vector perpendicular to two
given vectors.
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00:11:04 --> 00:11:08
We know two vectors in that
plane, for example,
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00:11:08 --> 00:11:11
P1P2, and P1P3.
Actually, I could have used
144
00:11:11 --> 00:11:14
another permutation of these
points, but, let's use this.
145
00:11:14 --> 00:11:18
So, if I want to find a vector
that's perpendicular to both
146
00:11:18 --> 00:11:22
P1P2 and P1P3 at the same time,
all I have to do is take their
147
00:11:22 --> 00:11:27
cross product.
So, how do we find a vector
148
00:11:27 --> 00:11:32
that's perpendicular to the
plane?
149
00:11:32 --> 00:11:46
The answer is just the cross
product P1P2 cross P1P3.
150
00:11:46 --> 00:11:49
Say you actually took the
points in a different order,
151
00:11:49 --> 00:11:52
and you took P1P3 x P1P2.
You would get,
152
00:11:52 --> 00:11:55
of course, the opposite vector.
That is fine.
153
00:11:55 --> 00:11:58
Any plane actually has
infinitely many normal vectors.
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00:11:58 --> 00:12:03
You can just multiply a normal
vector by any constant,
155
00:12:03 --> 00:12:07
you will still get a normal
vector.
156
00:12:07 --> 00:12:12
So, that's going to be one of
the main uses of dot product.
157
00:12:12 --> 00:12:16
When we know two vectors in a
plane, it lets us find the
158
00:12:16 --> 00:12:21
normal vector to the plane,
and that is what we need to
159
00:12:21 --> 00:12:26
find the equation.
Now, why is that the same as
160
00:12:26 --> 00:12:33
our first answer over there?
Well, the condition that we
161
00:12:33 --> 00:12:39
have is that P1P dot N should be
0.
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00:12:39 --> 00:12:48
And we said N is actually P1P2
cross P1P3.
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00:12:48 --> 00:12:51
So, this is what we want to be
zero.
164
00:12:51 --> 00:12:56
Now, if you remember,
a long time ago (that was
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00:12:56 --> 00:13:04
Friday) we've introduced this
thing and called it the triple
166
00:13:04 --> 00:13:07
product.
And what we've seen is that the
167
00:13:07 --> 00:13:10
triple product is the same thing
as the determinant.
168
00:13:10 --> 00:13:13
So, in fact,
these two ways of thinking,
169
00:13:13 --> 00:13:17
one saying that the box formed
by these three vectors should be
170
00:13:17 --> 00:13:21
flat and have volume zero,
and the other one saying that
171
00:13:21 --> 00:13:25
we can find a normal vector and
then express the condition that
172
00:13:25 --> 00:13:29
a vector is in the plane if it's
perpendicular to the normal
173
00:13:29 --> 00:13:31
vector,
are actually giving us the same
174
00:13:31 --> 00:13:32
formula in the end.
175
00:13:32 --> 00:13:41
176
00:13:41 --> 00:13:46
OK, any quick questions before
we move on?
177
00:13:46 --> 00:13:50
STUDENT QUESTION:
are those two equal only when P
178
00:13:50 --> 00:13:53
is in the plane,
or no matter where it is?
179
00:13:53 --> 00:13:57
So, these two quantities,
P1P dot the cross product,
180
00:13:57 --> 00:14:02
or the determinant of the three
vectors, are always equal to
181
00:14:02 --> 00:14:04
each other.
They are always the same.
182
00:14:04 --> 00:14:08
And now, if a point is not in
the plane, then their numerical
183
00:14:08 --> 00:14:13
value will be nonzero.
If P is in the plane,
184
00:14:13 --> 00:14:26
it will be zero.
OK, let's move on and talk a
185
00:14:26 --> 00:14:35
bit about matrices.
Probably some of you have
186
00:14:35 --> 00:14:38
learnt about matrices a little
bit in high school,
187
00:14:38 --> 00:14:42
but certainly not all of you.
So let me just introduce you to
188
00:14:42 --> 00:14:46
a little bit about matrices --
just enough for what we will
189
00:14:46 --> 00:14:51
need later on in this class.
If you want to know everything
190
00:14:51 --> 00:14:56
about the secret life of
matrices, then you should take
191
00:14:56 --> 00:14:59
18.06 someday.
OK, what's going to be our
192
00:14:59 --> 00:15:02
motivation for matrices?
Well, in life,
193
00:15:02 --> 00:15:07
a lot of things are related by
linear formulas.
194
00:15:07 --> 00:15:10
And, even if they are not,
maybe sometimes you can
195
00:15:10 --> 00:15:12
approximate them by linear
formulas.
196
00:15:12 --> 00:15:30
So, often, we have linear
relations between variables --
197
00:15:30 --> 00:15:47
for example, if we do a change
of coordinate systems.
198
00:15:47 --> 00:15:52
For example,
say that we are in space,
199
00:15:52 --> 00:15:58
and we have a point.
Its coordinates might be,
200
00:15:58 --> 00:16:02
let me call them x1,
x2, x3 in my initial coordinate
201
00:16:02 --> 00:16:04
system.
But then, maybe I need to
202
00:16:04 --> 00:16:07
actually switch to different
coordinates to better solve the
203
00:16:07 --> 00:16:09
problem because they're more
adapted to other things that
204
00:16:09 --> 00:16:13
we'll do in the problem.
And so I have other coordinates
205
00:16:13 --> 00:16:18
axes, and in these new
coordinates, P will have
206
00:16:18 --> 00:16:22
different coordinates -- let me
call them, say,
207
00:16:22 --> 00:16:25
u1, u2, u3.
And then, the relation between
208
00:16:25 --> 00:16:29
the old and the new coordinates
is going to be given by linear
209
00:16:29 --> 00:16:33
formulas -- at least if I choose
the same origin.
210
00:16:33 --> 00:16:38
Otherwise, there might be
constant terms,
211
00:16:38 --> 00:16:50
which I will not insist on.
Let me just give an example.
212
00:16:50 --> 00:16:58
For example,
maybe, let's say u1 could be 2
213
00:16:58 --> 00:17:08
x1 3 x2 3 x3.
u2 might be 2 x1 4 x2 5 x3.
214
00:17:08 --> 00:17:16
u3 might be x1 x2 2 x3.
Do not ask me where these
215
00:17:16 --> 00:17:18
numbers come from.
I just made them up,
216
00:17:18 --> 00:17:23
that's just an example of what
might happen.
217
00:17:23 --> 00:17:30
You can put here your favorite
numbers if you want.
218
00:17:30 --> 00:17:35
Now, in order to express this
kind of linear relation,
219
00:17:35 --> 00:17:39
we can use matrices.
A matrix is just a table with
220
00:17:39 --> 00:17:45
numbers in it.
And we can reformulate this in
221
00:17:45 --> 00:17:54
terms of matrix multiplication
or matrix product.
222
00:17:54 --> 00:18:04
So, instead of writing this,
I will write that the matrix
223
00:18:04 --> 00:18:11
|2,3, 3; 2,4,
5; 1,1, 2| times the vector
224
00:18:11 --> 00:18:16
***amp***lt;x1,
x2, x3> is equal to
225
00:18:16 --> 00:18:22
***amp***lt;u1,
u2, u3>.
226
00:18:22 --> 00:18:26
Hopefully you see that there is
the same information content on
227
00:18:26 --> 00:18:29
both sides.
I just need to explain to you
228
00:18:29 --> 00:18:35
what this way of multiplying
tables of numbers means.
229
00:18:35 --> 00:18:40
Well, what it means is really
that we'll have exactly these
230
00:18:40 --> 00:18:45
same quantities.
Let me just say that more
231
00:18:45 --> 00:18:49
symbolically:
so maybe this matrix could be
232
00:18:49 --> 00:18:56
called A, and this we could call
X, and this one we could call U.
233
00:18:56 --> 00:19:00
Then we'll say A times X equals
U, which is a lot shorter than
234
00:19:00 --> 00:19:03
that.
Of course, I need to tell you
235
00:19:03 --> 00:19:07
what A, X, and U are in terms of
their entries for you to get the
236
00:19:07 --> 00:19:11
formula.
But it's a convenient notation.
237
00:19:11 --> 00:19:17
So, what does it mean to do a
matrix product?
238
00:19:17 --> 00:19:30
The entries in the matrix
product are obtained by taking
239
00:19:30 --> 00:19:37
dot products.
Let's say we are doing the
240
00:19:37 --> 00:19:48
product AX.
We do a dot products between
241
00:19:48 --> 00:20:00
the rows of A and the columns of
X.
242
00:20:00 --> 00:20:07
Here, A is a 3x3 matrix -- that
just means there's three rows
243
00:20:07 --> 00:20:14
and three columns.
And X is a column vector,
244
00:20:14 --> 00:20:20
which we can think of as a 3x1
matrix.
245
00:20:20 --> 00:20:27
It has three rows and only one
column.
246
00:20:27 --> 00:20:31
Now, what can we do?
Well, I said we are going to do
247
00:20:31 --> 00:20:35
a dot product between a row of
A: 2,3, 3, and a column of X:
248
00:20:35 --> 00:20:38
x1, x2, x3.
That dot product will be two
249
00:20:38 --> 00:20:43
times x1 plus three times x2
plus three times x3.
250
00:20:43 --> 00:20:47
OK, it's exactly what we want
to set u1 equal to.
251
00:20:47 --> 00:20:51
Let's do the second one.
I take the second row of A:
252
00:20:51 --> 00:20:55
2,4, 5, and I do the dot
product with x1,
253
00:20:55 --> 00:20:59
x2, x3.
I will get two times x1 plus
254
00:20:59 --> 00:21:04
four times x2 plus five times
x3, which is u2.
255
00:21:04 --> 00:21:10
And, same thing with the third
one: one times x1 plus one times
256
00:21:10 --> 00:21:18
x2 plus two times x3.
So that's matrix multiplication.
257
00:21:18 --> 00:21:27
Let me restate things more
generally.
258
00:21:27 --> 00:21:33
If I want to find the entries
of a product of two matrices,
259
00:21:33 --> 00:21:38
A and B -- I'm saying matrices,
but of course they could be
260
00:21:38 --> 00:21:41
vectors.
Vectors are now a special case
261
00:21:41 --> 00:21:44
of matrices, just by taking a
matrix of width one.
262
00:21:44 --> 00:21:54
So, if I have my matrix A,
and I have my matrix B,
263
00:21:54 --> 00:22:01
then I will get the product,
AB.
264
00:22:01 --> 00:22:08
Let's say for example -- this
works in any size -- let's say
265
00:22:08 --> 00:22:13
that A is a 3x4 matrix.
So, it has three rows,
266
00:22:13 --> 00:22:15
four columns.
And, here, I'm not going to
267
00:22:15 --> 00:22:17
give you all the values because
I'm not going to compute
268
00:22:17 --> 00:22:19
everything.
It would take the rest of the
269
00:22:19 --> 00:22:23
lecture.
And let's say that B is maybe
270
00:22:23 --> 00:22:28
size 4x2.
So, it has two columns and four
271
00:22:28 --> 00:22:30
rows.
And, let's say,
272
00:22:30 --> 00:22:33
for example,
that we have the second column:
273
00:22:33 --> 00:22:36
0,3, 0,2.
So, in A times B,
274
00:22:36 --> 00:22:43
the entries should be the dot
products between these rows and
275
00:22:43 --> 00:22:46
these columns.
Here, we have two columns.
276
00:22:46 --> 00:22:49
Here, we have three rows.
So, we should get three times
277
00:22:49 --> 00:22:55
two different possibilities.
And so the answer will have
278
00:22:55 --> 00:22:59
size 3x2.
We cannot compute most of them,
279
00:22:59 --> 00:23:02
because I did not give you
numbers, but one of them we can
280
00:23:02 --> 00:23:04
compute.
We can compute the value that
281
00:23:04 --> 00:23:07
goes here, namely,
this one in the second column.
282
00:23:07 --> 00:23:13
So, I select the second column
of B, and I take the first row
283
00:23:13 --> 00:23:16
of A, and I find:
1 times 0: 0.
284
00:23:16 --> 00:23:20
2 times 3: 6,
plus 0, plus 8,
285
00:23:20 --> 00:23:28
should make 14.
So, this entry right here is 14.
286
00:23:28 --> 00:23:34
In fact, let me tell you about
another way to set it up so that
287
00:23:34 --> 00:23:38
you can remember more easily
what goes where.
288
00:23:38 --> 00:23:43
One way that you can set it up
is you can put A here.
289
00:23:43 --> 00:23:49
You can put B up here,
and then you will get the
290
00:23:49 --> 00:23:53
answer here.
And, if you want to find what
291
00:23:53 --> 00:23:57
goes in a given slot here,
then you just look to its left
292
00:23:57 --> 00:24:01
and you look above it,
and you do the dot product
293
00:24:01 --> 00:24:07
between these guys.
That's an easy way to remember.
294
00:24:07 --> 00:24:09
First of all,
it tells you what the size of
295
00:24:09 --> 00:24:11
the answer will be.
The size will be what fits
296
00:24:11 --> 00:24:14
nicely in this box:
it should have the same width
297
00:24:14 --> 00:24:18
as B and the same height as A.
And second, it tells you which
298
00:24:18 --> 00:24:22
dot product to compute for each
position.
299
00:24:22 --> 00:24:27
You just look at what's to the
left, and what's above the given
300
00:24:27 --> 00:24:29
position.
Now, there's a catch.
301
00:24:29 --> 00:24:32
Can we multiply anything by
anything?
302
00:24:32 --> 00:24:35
Well, no.
I wouldn't ask the question
303
00:24:35 --> 00:24:38
otherwise.
But anyway, to be able to do
304
00:24:38 --> 00:24:41
this dot product,
we need to have the same number
305
00:24:41 --> 00:24:45
of entries here and here.
Otherwise, we can't say "take
306
00:24:45 --> 00:24:46
this times that,
plus this times that,
307
00:24:46 --> 00:24:50
and so on" if we run out of
space on one of them before the
308
00:24:50 --> 00:24:57
other.
So, the condition -- and it's
309
00:24:57 --> 00:25:12
important, so let me write it in
red -- is that the width of A
310
00:25:12 --> 00:25:22
must equal the height of B.
(OK, it's a bit cluttered,
311
00:25:22 --> 00:25:28
but hopefully you can still see
what I'm writing.)
312
00:25:28 --> 00:25:31
OK, now we know how to multiply
matrices.
313
00:25:31 --> 00:25:38
314
00:25:38 --> 00:25:41
So, what does it mean to
multiply matrices?
315
00:25:41 --> 00:25:47
Of course, we've seen in this
example that we can use a matrix
316
00:25:47 --> 00:25:52
to tell us how to transform from
x's to u's.
317
00:25:52 --> 00:25:54
And, that's an example of
multiplication.
318
00:25:54 --> 00:25:58
But now, let's see that we have
two matrices like that telling
319
00:25:58 --> 00:26:01
us how to transform from
something to something else.
320
00:26:01 --> 00:26:02
What does it mean to multiply
them?
321
00:26:02 --> 00:26:11
322
00:26:11 --> 00:26:25
I claim that the product AB
represents doing first the
323
00:26:25 --> 00:26:36
transformation B,
then transformation A.
324
00:26:36 --> 00:26:37
That's a slightly
counterintuitive thing,
325
00:26:37 --> 00:26:40
because we are used to writing
things from left to right.
326
00:26:40 --> 00:26:43
Unfortunately,
with matrices,
327
00:26:43 --> 00:26:48
you multiply things from right
to left.
328
00:26:48 --> 00:26:51
If you think about it,
say you have two functions,
329
00:26:51 --> 00:26:55
f and g, and you write f(g(x)),
it really means you apply first
330
00:26:55 --> 00:26:59
g then f.
It works the same way as that.
331
00:26:59 --> 00:27:06
OK, so why is this?
Well, if I write AB times X
332
00:27:06 --> 00:27:12
where X is some vector that I
want to transform,
333
00:27:12 --> 00:27:16
it's the same as A times BX.
This property is called
334
00:27:16 --> 00:27:19
associativity.
And, it's a good property of
335
00:27:19 --> 00:27:23
well-behaved products -- not of
cross product,
336
00:27:23 --> 00:27:27
by the way.
Matrix product is associative.
337
00:27:27 --> 00:27:30
That means we can actually
think of a product ABX and
338
00:27:30 --> 00:27:32
multiply them in whichever order
we want.
339
00:27:32 --> 00:27:37
We can start with BX or we can
start with AB.
340
00:27:37 --> 00:27:43
So, now, BX means we apply the
transformation B to X.
341
00:27:43 --> 00:27:46
And then, multiplying by A
means we apply the
342
00:27:46 --> 00:27:49
transformation A.
So, we first apply B,
343
00:27:49 --> 00:27:58
then we apply A.
That's the same as applying AB
344
00:27:58 --> 00:28:05
all at once.
Another thing -- a warning:
345
00:28:05 --> 00:28:10
AB and BA are not the same
thing at all.
346
00:28:10 --> 00:28:13
You can probably see that
already from this
347
00:28:13 --> 00:28:18
interpretation.
It's not the same thing to
348
00:28:18 --> 00:28:24
convert oranges to bananas and
then to carrots,
349
00:28:24 --> 00:28:28
or vice versa.
Actually, even worse:
350
00:28:28 --> 00:28:31
this thing might not even be
well defined.
351
00:28:31 --> 00:28:38
If the width of A equals the
height of B, we can do this
352
00:28:38 --> 00:28:42
product.
But it's not clear that the
353
00:28:42 --> 00:28:47
width of B will equal the height
of A, which is what we would
354
00:28:47 --> 00:28:50
need for that one.
So, the size condition,
355
00:28:50 --> 00:28:53
to be able to do the product,
might not make sense -- maybe
356
00:28:53 --> 00:28:56
one of the products doesn't make
sense.
357
00:28:56 --> 00:29:01
Even if they both make sense,
they are usually completely
358
00:29:01 --> 00:29:07
different things.
The next thing I need to tell
359
00:29:07 --> 00:29:13
you about is something called
the identity matrix.
360
00:29:13 --> 00:29:17
The identity matrix is the
matrix that does nothing.
361
00:29:17 --> 00:29:19
What does it mean to do nothing?
I don't mean the matrix is zero.
362
00:29:19 --> 00:29:23
The matrix zero would take X
and would always give you back
363
00:29:23 --> 00:29:26
zero.
That's not a very interesting
364
00:29:26 --> 00:29:29
transformation.
What I mean is the guy that
365
00:29:29 --> 00:29:33
takes X and gives you X again.
It's called I,
366
00:29:33 --> 00:29:38
and it has the property that IX
equals X for all X.
367
00:29:38 --> 00:29:41
So, it's the transformation
from something to itself.
368
00:29:41 --> 00:29:44
It's the obvious transformation
-- called the identity
369
00:29:44 --> 00:29:48
transformation.
So, how do we write that as a
370
00:29:48 --> 00:29:51
matrix?
Well, actually there's an
371
00:29:51 --> 00:29:56
identity for each size because,
depending on whether X has two
372
00:29:56 --> 00:30:01
entries or ten entries,
the matrix I needs to have a
373
00:30:01 --> 00:30:05
different size.
For example,
374
00:30:05 --> 00:30:10
the identity matrix of size 3x3
has entries one,
375
00:30:10 --> 00:30:15
one, one on the diagonal,
and zero everywhere else.
376
00:30:15 --> 00:30:22
OK, let's check.
If we multiply this with a
377
00:30:22 --> 00:30:28
vector -- start thinking about
it.
378
00:30:28 --> 00:30:31
What happens when multiply this
with the vector X?
379
00:30:31 --> 00:31:00
380
00:31:00 --> 00:31:11
OK, so let's say I multiply the
matrix I with a vector x1,
381
00:31:11 --> 00:31:15
x2, x3.
What will the first entry be?
382
00:31:15 --> 00:31:19
It will be the dot product
between ***amp***lt;1,0,0> and
383
00:31:19 --> 00:31:23
***amp***lt;x1 x2 x3>.
This vector is i hat.
384
00:31:23 --> 00:31:27
If you do the dot product with
i hat, you will get the first
385
00:31:27 --> 00:31:32
component -- that will be x1.
One times x1 plus zero, zero.
386
00:31:32 --> 00:31:35
Similarly here,
if I do the dot product,
387
00:31:35 --> 00:31:40
I get zero plus x2 plus zero.
I get x2, and here I get x3.
388
00:31:40 --> 00:31:44
OK, it works.
Same thing if I put here a
389
00:31:44 --> 00:31:48
matrix: I will get back the same
matrix.
390
00:31:48 --> 00:31:58
In general, the identity matrix
in size n x n is an n x n matrix
391
00:31:58 --> 00:32:07
with ones on the diagonal,
and zeroes everywhere else.
392
00:32:07 --> 00:32:11
You just put 1 at every
diagonal position and 0
393
00:32:11 --> 00:32:13
elsewhere.
And then, you can see that if
394
00:32:13 --> 00:32:15
you multiply that by a vector,
you'll get the same vector
395
00:32:15 --> 00:32:15
back.
396
00:32:15 --> 00:32:29
397
00:32:29 --> 00:32:39
OK, let me give you another
example of a matrix.
398
00:32:39 --> 00:32:53
Let's say that in the plane we
look at the transformation that
399
00:32:53 --> 00:33:05
does rotation by 90°,
let's say, counterclockwise.
400
00:33:05 --> 00:33:11
I claim that this is given by
the matrix: |0,1;
401
00:33:11 --> 00:33:19
- 1,0|.
Let's try to see why that is
402
00:33:19 --> 00:33:25
the case.
Well, if I do R times i hat --
403
00:33:25 --> 00:33:29
if I apply that to the first
vector,
404
00:33:29 --> 00:33:35
i hat: i hat will be
***amp***lt;1,0> so in this
405
00:33:35 --> 00:33:39
product,
first you will get 0,
406
00:33:39 --> 00:33:46
and then you will get 1.
You get j hat.
407
00:33:46 --> 00:33:53
OK, so this thing sends i hat
to j hat.
408
00:33:53 --> 00:34:06
What about j hat?
Well, you get negative one.
409
00:34:06 --> 00:34:10
And then you get 0.
So, that's minus i hat.
410
00:34:10 --> 00:34:15
So, j is sent towards here.
And, in general,
411
00:34:15 --> 00:34:19
if you apply it to a vector
with components x,y,
412
00:34:19 --> 00:34:29
then you will get back -y,x,
which is the formula we've seen
413
00:34:29 --> 00:34:39
for rotating a vector by 90°.
So, it seems to do what we want.
414
00:34:39 --> 00:34:47
By the way, the columns in this
matrix represent what happens to
415
00:34:47 --> 00:34:53
each basis vector,
to the vectors i and j.
416
00:34:53 --> 00:34:57
This guy here is exactly what
we get when we multiply R by i.
417
00:34:57 --> 00:35:05
And, when we multiply R by j,
we get this guy here.
418
00:35:05 --> 00:35:08
So, what's interesting about
this matrix?
419
00:35:08 --> 00:35:12
Well, we can do computations
with matrices in ways that are
420
00:35:12 --> 00:35:15
easier than writing coordinate
change formulas.
421
00:35:15 --> 00:35:19
For example,
if you compute R squared,
422
00:35:19 --> 00:35:23
so if you multiply R with
itself: I'll let you do it as an
423
00:35:23 --> 00:35:28
exercise,
but you will find that you get
424
00:35:28 --> 00:35:33
|-1,0;0,-1|.
So, that's minus the identity
425
00:35:33 --> 00:35:35
matrix.
Why is that?
426
00:35:35 --> 00:35:39
Well, if I rotate something by
90° and then I rotate by 90°
427
00:35:39 --> 00:35:42
again, then I will rotate by
180�.
428
00:35:42 --> 00:35:46
That means I will actually just
go to the opposite point around
429
00:35:46 --> 00:35:51
the origin.
So, I will take (x,y) to
430
00:35:51 --> 00:35:58
(-x,-y).
And if I applied R four times,
431
00:35:58 --> 00:36:06
R^4 would be identity.
OK, questions?
432
00:36:06 --> 00:36:11
STUDENT QUESTION:
when you said R equals that
433
00:36:11 --> 00:36:14
matrix, is that the definition
of R?
434
00:36:14 --> 00:36:17
How did I come up with this R?
Well, secretly,
435
00:36:17 --> 00:36:21
I worked pretty hard to find
the entries that would tell me
436
00:36:21 --> 00:36:25
how to rotate something by 90°
counterclockwise.
437
00:36:25 --> 00:36:32
So, remember:
what we saw last time or in the
438
00:36:32 --> 00:36:39
first lecture is that,
to rotate a vector by 90°,
439
00:36:39 --> 00:36:46
we should change (x,
y) to (-y, x).
440
00:36:46 --> 00:36:52
And now I'm trying to express
this transformation as a matrix.
441
00:36:52 --> 00:36:57
So, maybe you can call these
guys u and v,
442
00:36:57 --> 00:37:02
and then you write that u
equals 0x-1y,
443
00:37:02 --> 00:37:08
and that v equals 1x 0y.
So that's how I would find it.
444
00:37:08 --> 00:37:13
Here, I just gave it to you
already made,
445
00:37:13 --> 00:37:19
so you didn't really see how I
found it.
446
00:37:19 --> 00:37:30
You will see more about
rotations on the problem set.
447
00:37:30 --> 00:37:35
OK, next I need to tell you how
to invert matrices.
448
00:37:35 --> 00:37:39
So, what's the point of
matrices?
449
00:37:39 --> 00:37:41
It's that it gives us a nice
way to think about changes of
450
00:37:41 --> 00:37:43
variables.
And, in particular,
451
00:37:43 --> 00:37:48
if we know how to express U in
terms of X, maybe we'd like to
452
00:37:48 --> 00:37:51
know how to express X in terms
of U.
453
00:37:51 --> 00:37:54
Well, we can do that,
because we've learned how to
454
00:37:54 --> 00:37:58
solve linear systems like this.
So in principle,
455
00:37:58 --> 00:38:01
we could start working,
substituting and so on,
456
00:38:01 --> 00:38:06
to find formulas for x1,
x2, x3 as functions of u1,
457
00:38:06 --> 00:38:09
u2, u3.
And the relation will be,
458
00:38:09 --> 00:38:11
again, a linear relation.
It will, again,
459
00:38:11 --> 00:38:14
be given by a matrix.
Well, what's that matrix?
460
00:38:14 --> 00:38:17
It's the inverse
transformation.
461
00:38:17 --> 00:38:21
It's the inverse of the matrix
A.
462
00:38:21 --> 00:38:24
So, we need to learn how to
find the inverse of a matrix
463
00:38:24 --> 00:38:25
directly.
464
00:38:25 --> 00:38:43
465
00:38:43 --> 00:38:48
The inverse of A,
by definition,
466
00:38:48 --> 00:38:56
is a matrix M,
with the property that if I
467
00:38:56 --> 00:39:03
multiply A by M,
then I get identity.
468
00:39:03 --> 00:39:07
And, if I multiply M by A,
I also get identity.
469
00:39:07 --> 00:39:10
The two properties are
equivalent.
470
00:39:10 --> 00:39:13
That means, if I apply first
the transformation A,
471
00:39:13 --> 00:39:16
then the transformation M,
actually I undo the
472
00:39:16 --> 00:39:18
transformation A,
and vice versa.
473
00:39:18 --> 00:39:24
These two transformations are
the opposite of each other,
474
00:39:24 --> 00:39:28
or I should say the inverse of
each other.
475
00:39:28 --> 00:39:37
For this to make sense,
we need A to be a square
476
00:39:37 --> 00:39:41
matrix.
It must have size n by n.
477
00:39:41 --> 00:39:45
It can be any size,
but it must have the same
478
00:39:45 --> 00:39:50
number of rows as columns.
It's a general fact that you
479
00:39:50 --> 00:39:55
will see more in detail in
linear algebra if you take it.
480
00:39:55 --> 00:40:09
Let's just admit it.
The matrix M will be denoted by
481
00:40:09 --> 00:40:13
A inverse.
Then, what is it good for?
482
00:40:13 --> 00:40:18
Well, for example,
finding the solution to a
483
00:40:18 --> 00:40:21
linear system.
What's a linear system in our
484
00:40:21 --> 00:40:24
new language?
It's: a matrix times some
485
00:40:24 --> 00:40:28
unknown vector,
X, equals some known vector,
486
00:40:28 --> 00:40:32
B.
How do we solve that?
487
00:40:32 --> 00:40:37
We just compute:
X equals A inverse B.
488
00:40:37 --> 00:40:42
Why does that work?
How do I get from here to here?
489
00:40:42 --> 00:40:43
Let's be careful.
490
00:40:43 --> 00:40:51
491
00:40:51 --> 00:40:54
(I'm going to reuse this
matrix, but I'm going to erase
492
00:40:54 --> 00:40:57
it nonetheless and I'll just
rewrite it).
493
00:40:57 --> 00:41:21
494
00:41:21 --> 00:41:30
If AX=B, then let's multiply
both sides by A inverse.
495
00:41:30 --> 00:41:35
A inverse times AX is A inverse
B.
496
00:41:35 --> 00:41:41
And then, A inverse times A is
identity, so I get:
497
00:41:41 --> 00:41:46
X equals A inverse B.
That's how I solved my system
498
00:41:46 --> 00:41:48
of equations.
So, if you have a calculator
499
00:41:48 --> 00:41:51
that can invert matrices,
then you can solve linear
500
00:41:51 --> 00:41:55
systems very quickly.
Now, we should still learn how
501
00:41:55 --> 00:41:58
to compute these things.
Yes?
502
00:41:58 --> 00:42:03
[Student Questions:]"How do you
know that A inverse will be on
503
00:42:03 --> 00:42:07
the left of B and not after it "
Well,
504
00:42:07 --> 00:42:10
it's exactly this derivation.
So, if you are not sure,
505
00:42:10 --> 00:42:13
then just reproduce this
calculation.
506
00:42:13 --> 00:42:16
To get from here to here,
what I did is I multiplied
507
00:42:16 --> 00:42:20
things on the left by A inverse,
and then this guy simplify.
508
00:42:20 --> 00:42:23
If I had put A inverse on the
right, I would have AX A
509
00:42:23 --> 00:42:27
inverse, which might not make
sense, and even if it makes
510
00:42:27 --> 00:42:31
sense, it doesn't simplify.
So, the basic rule is that you
511
00:42:31 --> 00:42:35
have to multiply by A inverse on
the left so that it cancels with
512
00:42:35 --> 00:42:38
this A that's on the left.
STUDENT QUESTION:
513
00:42:38 --> 00:42:41
"And if you put it on the left
on this side then it will be on
514
00:42:41 --> 00:42:43
the left with B as well?" That's
correct,
515
00:42:43 --> 00:42:46
in our usual way of dealing
with matrices,
516
00:42:46 --> 00:42:49
where the vectors are column
vectors.
517
00:42:49 --> 00:42:52
It's just something to
remember: if you have a square
518
00:42:52 --> 00:42:56
matrix times a column vector,
the product that makes sense is
519
00:42:56 --> 00:42:58
with the matrix on the left,
and the vector on the right.
520
00:42:58 --> 00:43:04
The other one just doesn't work.
You cannot take X times A if A
521
00:43:04 --> 00:43:11
is a square matrix and X is a
column vector.
522
00:43:11 --> 00:43:16
This product AX makes sense.
The other one XA doesn't make
523
00:43:16 --> 00:43:19
sense.
It's not the right size.
524
00:43:19 --> 00:43:23
OK.
What we need to do is to learn
525
00:43:23 --> 00:43:29
how to invert a matrix.
It's a useful thing to know,
526
00:43:29 --> 00:43:32
first for your general
knowledge, and second because
527
00:43:32 --> 00:43:38
it's actually useful for things
we'll see later in this class.
528
00:43:38 --> 00:43:40
In particular,
on the exam,
529
00:43:40 --> 00:43:45
you will need to know how to
invert a matrix by hand.
530
00:43:45 --> 00:43:50
This formula is actually good
for small matrices,
531
00:43:50 --> 00:43:52
3x3,4x4.
It's not good at all if you
532
00:43:52 --> 00:43:54
have a matrix of size
1,000x1,000.
533
00:43:54 --> 00:43:59
So, in computer software,
actually for small matrices
534
00:43:59 --> 00:44:02
they do this,
but for larger matrices,
535
00:44:02 --> 00:44:09
they use other algorithms.
Let's just see how we do it.
536
00:44:09 --> 00:44:13
First of all I will give you
the final answer.
537
00:44:13 --> 00:44:19
And of course I will need to
explain what the answer means.
538
00:44:19 --> 00:44:22
We will have to compute
something called the adjoint
539
00:44:22 --> 00:44:24
matrix.
I will tell you how to do that.
540
00:44:24 --> 00:44:35
And then, we will divide by the
determinant of A.
541
00:44:35 --> 00:44:38
How do we get to the adjoint
matrix?
542
00:44:38 --> 00:44:46
Let's go through the steps on a
3x3 example -- the steps are the
543
00:44:46 --> 00:44:52
same no matter what the size is,
but let's do 3x3.
544
00:44:52 --> 00:44:56
So, let's say that I'm giving
you the matrix A -- let's say
545
00:44:56 --> 00:44:59
it's the same as the one that I
erased earlier.
546
00:44:59 --> 00:45:08
That was the one relating our
X's and our U's.
547
00:45:08 --> 00:45:18
The first thing I want to do is
find something called the
548
00:45:18 --> 00:45:22
minors.
What's a minor?
549
00:45:22 --> 00:45:24
It's a slightly smaller
determinant.
550
00:45:24 --> 00:45:28
We've already seen them without
calling them that way.
551
00:45:28 --> 00:45:32
The matrix of minors will have
again the same size.
552
00:45:32 --> 00:45:37
Let's say we want this entry.
Then, we just delete this row
553
00:45:37 --> 00:45:40
and this column,
and we are left with a 2x2
554
00:45:40 --> 00:45:44
determinant.
So, here, we'll put the
555
00:45:44 --> 00:45:49
determinant 4,5,
1,2, which is 4 times 2:
556
00:45:49 --> 00:45:51
8 -- minus 5:
3.
557
00:45:51 --> 00:45:53
Let's do the next one.
So, for this entry,
558
00:45:53 --> 00:45:55
I'll delete this row and this
column.
559
00:45:55 --> 00:46:00
I'm left with 2,5, 1,2.
The determinant will be 2 times
560
00:46:00 --> 00:46:04
2 minus 5, which is negative 1.
Then minus 2,
561
00:46:04 --> 00:46:09
then I get to the second row,
so I get to this entry.
562
00:46:09 --> 00:46:12
To find the minor here,
I will delete this row and this
563
00:46:12 --> 00:46:15
column.
And I'm left with 3,3, 1,2.
564
00:46:15 --> 00:46:24
3 times 2 minus 3 is 3.
Let me just cheat and give you
565
00:46:24 --> 00:46:31
the others -- I think I've shown
you that I can do them.
566
00:46:31 --> 00:46:34
Let's just explain the last one
again.
567
00:46:34 --> 00:46:37
The last one is 2.
To find the minor here,
568
00:46:37 --> 00:46:41
I delete this column and this
row, and I take this
569
00:46:41 --> 00:46:44
determinant: 2 times 4 minus 2
times 3.
570
00:46:44 --> 00:46:49
So it's the same kind of
manipulation that we've seen
571
00:46:49 --> 00:46:53
when we've taken determinants
and cross products.
572
00:46:53 --> 00:46:59
Step two: we go to another
matrix that's called cofactors.
573
00:46:59 --> 00:47:03
So, the cofactors are pretty
much the same thing as the
574
00:47:03 --> 00:47:07
minors except the signs are
slightly different.
575
00:47:07 --> 00:47:16
What we do is that we flip
signs according to a
576
00:47:16 --> 00:47:22
checkerboard diagram.
You start with a plus in the
577
00:47:22 --> 00:47:26
upper left corner,
and you alternate pluses and
578
00:47:26 --> 00:47:28
minuses.
The rule is:
579
00:47:28 --> 00:47:33
if there is a plus somewhere,
then there's a minus next to it
580
00:47:33 --> 00:47:36
and below it.
And then, below a minus or to
581
00:47:36 --> 00:47:38
the right of a minus,
there's a plus.
582
00:47:38 --> 00:47:43
So that's how it looks in size
3x3.
583
00:47:43 --> 00:47:46
What do I mean by that?
I don't mean,
584
00:47:46 --> 00:47:48
make this positive,
make this negative,
585
00:47:48 --> 00:47:50
and so on.
That's not what I mean.
586
00:47:50 --> 00:47:53
What I mean is:
if there's a plus,
587
00:47:53 --> 00:47:59
that means leave it alone -- we
don't do anything to it.
588
00:47:59 --> 00:48:05
If there's a minus,
that means we flip the sign.
589
00:48:05 --> 00:48:17
So, here, we'd get:
3, then 1, -2,
590
00:48:17 --> 00:48:25
-3,1, 1...
3,-4, and 2.
591
00:48:25 --> 00:48:29
OK, that step is pretty easy.
The only hard step in terms of
592
00:48:29 --> 00:48:32
calculations is the first one
because you have to compute all
593
00:48:32 --> 00:48:33
of these 2x2 determinants.
594
00:48:33 --> 00:48:40
595
00:48:40 --> 00:48:44
By the way, this minus sign
here is actually related to the
596
00:48:44 --> 00:48:47
way in which,
when we do a cross product,
597
00:48:47 --> 00:48:51
we have a minus sign for the
second entry.
598
00:48:51 --> 00:49:00
OK, we're almost done.
The third step is to transpose.
599
00:49:00 --> 00:49:03
What does it mean to transpose?
It means: you read the rows of
600
00:49:03 --> 00:49:07
your matrix and write them as
columns, or vice versa.
601
00:49:07 --> 00:49:16
So we switch rows and columns.
What do we get?
602
00:49:16 --> 00:49:19
Well, let's just read the
matrix horizontally and write it
603
00:49:19 --> 00:49:24
vertically.
We read 3,1, - 2: 3,1, - 2.
604
00:49:24 --> 00:49:29
Then we read -3 3,1,
1: - 3,1, 1.
605
00:49:29 --> 00:49:39
Then, 3, - 4,2: 3, - 4,2.
That's pretty easy.
606
00:49:39 --> 00:49:44
We're almost done.
What we get here is this is the
607
00:49:44 --> 00:49:52
adjoint matrix.
So, the fourth and last step is
608
00:49:52 --> 00:49:58
to divide by the determinant of
A.
609
00:49:58 --> 00:50:04
We have to compute the
determinant -- the determinant
610
00:50:04 --> 00:50:08
of A, not the determinant of
this guy.
611
00:50:08 --> 00:50:16
So: 2,3, 3,2, 4,5, 1,1, 2.
I'll let you check my
612
00:50:16 --> 00:50:21
computation.
I found that it's equal to 3.
613
00:50:21 --> 00:50:30
So the final answer is that A
inverse is one third of the
614
00:50:30 --> 00:50:35
matrix we got there:
|3, - 3,3, 1,1,
615
00:50:35 --> 00:50:39
- 4, - 2,1, 2|.
Now, remember,
616
00:50:39 --> 00:50:43
A told us how to find the u's
in terms of the x's.
617
00:50:43 --> 00:50:47
This tells us how to find x-s
in terms of u-s:
618
00:50:47 --> 00:50:52
if you multiply x1,x2,x3 by
this you get u1,u2,u3.
619
00:50:52 --> 00:50:56
It also tells you how to solve
a linear system:
620
00:50:56 --> 00:51:03
A times X equals something.
621
00:51:03 --> 00:51:08