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Remember last time -- -- we
learned about the cross product
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00:00:34,000 --> 00:00:42,000
of vectors in space.
Remember the definition of
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00:00:42,000 --> 00:00:48,000
cross product is in terms of
this determinant det| i hat,
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00:00:48,000 --> 00:00:53,000
j hat, k hat,
and then the components of A,
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00:00:53,000 --> 00:00:57,000
a1, a2, a3,
and then the components of B|
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00:00:57,000 --> 00:01:02,000
This is not an actual
determinant because these are
13
00:01:02,000 --> 00:01:05,000
not numbers.
But it's a symbolic notation,
14
00:01:05,000 --> 00:01:08,000
to remember what the actual
formula is.
15
00:01:08,000 --> 00:01:12,000
The actual formula is obtained
by expanding the determinant.
16
00:01:12,000 --> 00:01:19,000
So, we actually get the
determinant of a2,
17
00:01:19,000 --> 00:01:27,000
a3, b2, b3 times i hat,
minus the determinant of a1,
18
00:01:27,000 --> 00:01:35,000
a3, b1, b3 times j hat plus the
determinant of a1,
19
00:01:35,000 --> 00:01:42,000
a2, b1, b2, times k hat.
And we also saw a more
20
00:01:42,000 --> 00:01:47,000
geometric definition of the
cross product.
21
00:01:47,000 --> 00:01:56,000
We have learned that the length
of the cross product is equal to
22
00:01:56,000 --> 00:02:04,000
the area of the parallelogram
with sides A and B.
23
00:02:17,000 --> 00:02:26,000
We have also learned that the
direction of the cross product
24
00:02:26,000 --> 00:02:37,000
is given by taking the direction
that's perpendicular to A and B.
25
00:02:37,000 --> 00:02:42,000
If I draw A and B in a plane
(they determine a plane),
26
00:02:42,000 --> 00:02:48,000
then the cross product should
go in the direction that's
27
00:02:48,000 --> 00:02:53,000
perpendicular to that plane.
Now, there's two different
28
00:02:53,000 --> 00:02:56,000
possible directions that are
perpendicular to a plane.
29
00:02:56,000 --> 00:03:04,000
And, to decide which one it is,
we use the right-hand rule,
30
00:03:04,000 --> 00:03:07,000
which says if you extend your
right hand in the direction of
31
00:03:07,000 --> 00:03:10,000
the vector A,
then curve your fingers in the
32
00:03:10,000 --> 00:03:14,000
direction of B,
then your thumb will go in the
33
00:03:14,000 --> 00:03:20,000
direction of the cross product.
One thing I didn't quite get to
34
00:03:20,000 --> 00:03:26,000
say last time is that there are
some funny manipulation rules.
35
00:03:26,000 --> 00:03:29,000
What are we allowed to do or
not do with cross products?
36
00:03:29,000 --> 00:03:35,000
So, let me tell you right away
the most surprising one if
37
00:03:35,000 --> 00:03:41,000
you've never seen it before:
A cross B and B cross A are not
38
00:03:41,000 --> 00:03:45,000
the same thing.
Why are they not the same thing?
39
00:03:45,000 --> 00:03:49,000
Well, one way to see it is to
think geometrically.
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00:03:49,000 --> 00:03:52,000
The parallelogram still has the
same area, and it's still in the
41
00:03:52,000 --> 00:03:54,000
same plane.
So, the cross product is still
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00:03:54,000 --> 00:03:58,000
perpendicular to the same plane.
But, what happens is that,
43
00:03:58,000 --> 00:04:01,000
if you try to apply the
right-hand rule but exchange the
44
00:04:01,000 --> 00:04:04,000
roles of A and B,
then you will either injure
45
00:04:04,000 --> 00:04:06,000
yourself,
or your thumb will end up
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00:04:06,000 --> 00:04:08,000
pointing in the opposite
direction.
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00:04:08,000 --> 00:04:12,000
So, in fact,
B cross A and A cross B are
48
00:04:12,000 --> 00:04:17,000
opposite of each other.
And you can check that in the
49
00:04:17,000 --> 00:04:19,000
formula because,
for example,
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00:04:19,000 --> 00:04:22,000
the i component is a2 b3 minus
a3 b2.
51
00:04:22,000 --> 00:04:27,000
If you swap the roles of A and
B, you will also have to change
52
00:04:27,000 --> 00:04:30,000
the signs.
That's a slightly surprising
53
00:04:30,000 --> 00:04:33,000
thing, but you will see one
easily adjusts to it.
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00:04:33,000 --> 00:04:36,000
It just means one must resist
the temptation to write AxB
55
00:04:36,000 --> 00:04:40,000
equals BxA.
Whenever you do that,
56
00:04:40,000 --> 00:04:45,000
put a minus sign.
Now, in particular,
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00:04:45,000 --> 00:04:53,000
what happens if I do A cross A?
Well, I will get zero.
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00:04:53,000 --> 00:04:54,000
And, there's many ways to see
that.
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00:04:54,000 --> 00:04:58,000
One is to use the formula.
Also, you can see that the
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00:04:58,000 --> 00:05:02,000
parallelogram formed by A and A
is completely flat,
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00:05:02,000 --> 00:05:06,000
and it has area zero.
So, we get the zero vector.
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00:05:17,000 --> 00:05:20,000
Hopefully you got practice with
cross products,
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00:05:20,000 --> 00:05:23,000
and computing them,
in recitation yesterday.
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00:05:23,000 --> 00:05:29,000
Let me just point out one
important application of cross
65
00:05:29,000 --> 00:05:33,000
product that maybe you haven't
seen yet.
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00:05:33,000 --> 00:05:36,000
Let's say that I'm given three
points in space,
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00:05:36,000 --> 00:05:39,000
and I want to find the equation
of the plane that contains them.
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00:05:39,000 --> 00:05:45,000
So, say I have P1,
P2, P3, three points in space.
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00:05:45,000 --> 00:05:48,000
They determine a plane,
at least if they are not
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00:05:48,000 --> 00:05:51,000
aligned, and we would like to
find the equation of the plane
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00:05:51,000 --> 00:05:56,000
that they determine.
That means, let's say that we
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00:05:56,000 --> 00:06:01,000
have a point,
P, in space with coordinates x,
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00:06:01,000 --> 00:06:07,000
y, z.
Well, to find the equation of
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00:06:07,000 --> 00:06:14,000
the plane -- -- the plane
containing P1,
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00:06:14,000 --> 00:06:22,000
P2, and P3,
we need to find a condition on
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00:06:22,000 --> 00:06:26,000
the coordinates x,
y, z,
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00:06:26,000 --> 00:06:41,000
telling us whether P is in the
plane or not.
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00:06:41,000 --> 00:06:44,000
We have several ways of doing
that.
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00:06:44,000 --> 00:06:47,000
For example,
one thing we could do.
80
00:06:47,000 --> 00:06:51,000
Let me just backtrack to
determinants that we saw last
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00:06:51,000 --> 00:06:56,000
time.
One way to think about it is to
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00:06:56,000 --> 00:07:03,000
consider these vectors,
P1P2, P1P3, and P1P.
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00:07:03,000 --> 00:07:07,000
The question of whether they
are all in the same plane is the
84
00:07:07,000 --> 00:07:12,000
same as asking ourselves whether
the parallelepiped that they
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00:07:12,000 --> 00:07:15,000
form is actually completely
flattened.
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00:07:15,000 --> 00:07:18,000
So, if I try to form a
parallelepiped with these three
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00:07:18,000 --> 00:07:21,000
sides, and P is not in the
plane, then it will have some
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00:07:21,000 --> 00:07:24,000
volume.
But, if P is in the plane,
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00:07:24,000 --> 00:07:26,000
then it's actually completely
squished.
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00:07:26,000 --> 00:07:31,000
So,one possible answer,
one possible way to think of
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00:07:31,000 --> 00:07:37,000
the equation of a plane is that
the determinant of these vectors
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00:07:37,000 --> 00:07:42,000
should be zero.
Take the determinant of (vector
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00:07:42,000 --> 00:07:48,000
P1P,vector P1P2,vector P1P3)
equals 0 (if you do it in a
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00:07:48,000 --> 00:07:53,000
different order it doesn't
really matter).
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00:07:53,000 --> 00:07:58,000
One possible way to express the
condition that P is in the plane
96
00:07:58,000 --> 00:08:02,000
is to say that the determinant
of these three vectors has to be
97
00:08:02,000 --> 00:08:05,000
zero.
And, if I am given coordinates
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00:08:05,000 --> 00:08:07,000
for these points -- I'm not
giving you numbers,
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00:08:07,000 --> 00:08:10,000
but if I gave you numbers,
then you would be able to plug
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00:08:10,000 --> 00:08:14,000
those numbers in.
So, you could compute these two
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00:08:14,000 --> 00:08:16,000
vectors P1P2 and P1P3
explicitly.
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00:08:16,000 --> 00:08:19,000
But, of course,
P1P would depend on x,
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00:08:19,000 --> 00:08:21,000
y, and z.
So, when you compute the
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00:08:21,000 --> 00:08:24,000
determinant, you get a formula
that involves x,
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00:08:24,000 --> 00:08:26,000
y, and z.
And you'll find that this
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00:08:26,000 --> 00:08:29,000
condition on x,
y, z is the equation of a
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00:08:29,000 --> 00:08:32,000
plane.
We're going to see more about
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00:08:32,000 --> 00:08:36,000
that pretty soon.
Now, let me tell you a slightly
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00:08:36,000 --> 00:08:40,000
faster way of doing it.
Actually, it's not much faster,
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00:08:40,000 --> 00:08:44,000
It's pretty much the same
calculation, but it's maybe more
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00:08:44,000 --> 00:08:50,000
enlightening.
Let me actually show you a nice
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00:08:50,000 --> 00:08:56,000
color picture that I prepared
for this.
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00:08:56,000 --> 00:09:00,000
One thing that's on this
picture that I haven't drawn
114
00:09:00,000 --> 00:09:02,000
before is the normal vector to
the plane.
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00:09:02,000 --> 00:09:06,000
Why is that?
Well, let's say that we know
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00:09:06,000 --> 00:09:09,000
how to find a vector that's
perpendicular to our plane.
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00:09:09,000 --> 00:09:13,000
Then, what does it mean for the
point, P, to be in the plane?
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00:09:13,000 --> 00:09:19,000
It means that the direction
from P1 to P has to be
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00:09:19,000 --> 00:09:29,000
perpendicular to this vector N.
So here's another solution:
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00:09:29,000 --> 00:09:43,000
P is in the plane exactly when
the vector P1P is perpendicular
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00:09:43,000 --> 00:09:48,000
to N,
where N is some vector that's
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00:09:48,000 --> 00:10:05,000
perpendicular to the plane.
N is called a normal vector.
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00:10:05,000 --> 00:10:08,000
How do we rephrase this
condition?
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00:10:08,000 --> 00:10:13,000
Well, we've learned how to
detect whether two vectors are
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00:10:13,000 --> 00:10:18,000
perpendicular to each other
using dot product (that was the
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00:10:18,000 --> 00:10:21,000
first lecture).
These two vectors are
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00:10:21,000 --> 00:10:25,000
perpendicular exactly when their
dot product is zero.
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00:10:25,000 --> 00:10:32,000
So, concretely,
if we have a point P1 given to
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00:10:32,000 --> 00:10:34,000
us,
and say we have been able to
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00:10:34,000 --> 00:10:37,000
compute the vector N,
then when we actually compute
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00:10:37,000 --> 00:10:40,000
what happens,
here we will have the
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00:10:40,000 --> 00:10:41,000
coordinates x,
y, z, of a point P,
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00:10:41,000 --> 00:10:44,000
and we will get some condition
on x, y, z.
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00:10:44,000 --> 00:10:47,000
That will be the equation of a
plane.
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00:10:47,000 --> 00:10:50,000
Now, why are these things the
same?
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00:10:50,000 --> 00:10:54,000
Well, before I can tell you
that, I should tell you how to
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00:10:54,000 --> 00:10:57,000
find a normal vector.
Maybe you are already starting
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00:10:57,000 --> 00:11:01,000
to see what the method should
be, because we know how to find
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00:11:01,000 --> 00:11:04,000
a vector perpendicular to two
given vectors.
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00:11:04,000 --> 00:11:08,000
We know two vectors in that
plane, for example,
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00:11:08,000 --> 00:11:11,000
P1P2, and P1P3.
Actually, I could have used
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00:11:11,000 --> 00:11:14,000
another permutation of these
points, but, let's use this.
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00:11:14,000 --> 00:11:18,000
So, if I want to find a vector
that's perpendicular to both
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00:11:18,000 --> 00:11:22,000
P1P2 and P1P3 at the same time,
all I have to do is take their
145
00:11:22,000 --> 00:11:27,000
cross product.
So, how do we find a vector
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00:11:27,000 --> 00:11:32,000
that's perpendicular to the
plane?
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00:11:32,000 --> 00:11:46,000
The answer is just the cross
product P1P2 cross P1P3.
148
00:11:46,000 --> 00:11:49,000
Say you actually took the
points in a different order,
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00:11:49,000 --> 00:11:52,000
and you took P1P3 x P1P2.
You would get,
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00:11:52,000 --> 00:11:55,000
of course, the opposite vector.
That is fine.
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00:11:55,000 --> 00:11:58,000
Any plane actually has
infinitely many normal vectors.
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00:11:58,000 --> 00:12:03,000
You can just multiply a normal
vector by any constant,
153
00:12:03,000 --> 00:12:07,000
you will still get a normal
vector.
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00:12:07,000 --> 00:12:12,000
So, that's going to be one of
the main uses of dot product.
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00:12:12,000 --> 00:12:16,000
When we know two vectors in a
plane, it lets us find the
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00:12:16,000 --> 00:12:21,000
normal vector to the plane,
and that is what we need to
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00:12:21,000 --> 00:12:26,000
find the equation.
Now, why is that the same as
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00:12:26,000 --> 00:12:33,000
our first answer over there?
Well, the condition that we
159
00:12:33,000 --> 00:12:39,000
have is that P1P dot N should be
0.
160
00:12:39,000 --> 00:12:48,000
And we said N is actually P1P2
cross P1P3.
161
00:12:48,000 --> 00:12:51,000
So, this is what we want to be
zero.
162
00:12:51,000 --> 00:12:56,000
Now, if you remember,
a long time ago (that was
163
00:12:56,000 --> 00:13:04,000
Friday) we've introduced this
thing and called it the triple
164
00:13:04,000 --> 00:13:07,000
product.
And what we've seen is that the
165
00:13:07,000 --> 00:13:10,000
triple product is the same thing
as the determinant.
166
00:13:10,000 --> 00:13:13,000
So, in fact,
these two ways of thinking,
167
00:13:13,000 --> 00:13:17,000
one saying that the box formed
by these three vectors should be
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00:13:17,000 --> 00:13:21,000
flat and have volume zero,
and the other one saying that
169
00:13:21,000 --> 00:13:25,000
we can find a normal vector and
then express the condition that
170
00:13:25,000 --> 00:13:29,000
a vector is in the plane if it's
perpendicular to the normal
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00:13:29,000 --> 00:13:31,000
vector,
are actually giving us the same
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00:13:31,000 --> 00:13:32,000
formula in the end.
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00:13:41,000 --> 00:13:46,000
OK, any quick questions before
we move on?
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00:13:46,000 --> 00:13:50,000
STUDENT QUESTION:
are those two equal only when P
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00:13:50,000 --> 00:13:53,000
is in the plane,
or no matter where it is?
176
00:13:53,000 --> 00:13:57,000
So, these two quantities,
P1P dot the cross product,
177
00:13:57,000 --> 00:14:02,000
or the determinant of the three
vectors, are always equal to
178
00:14:02,000 --> 00:14:04,000
each other.
They are always the same.
179
00:14:04,000 --> 00:14:08,000
And now, if a point is not in
the plane, then their numerical
180
00:14:08,000 --> 00:14:13,000
value will be nonzero.
If P is in the plane,
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00:14:13,000 --> 00:14:26,000
it will be zero.
OK, let's move on and talk a
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00:14:26,000 --> 00:14:35,000
bit about matrices.
Probably some of you have
183
00:14:35,000 --> 00:14:38,000
learnt about matrices a little
bit in high school,
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00:14:38,000 --> 00:14:42,000
but certainly not all of you.
So let me just introduce you to
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00:14:42,000 --> 00:14:46,000
a little bit about matrices --
just enough for what we will
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00:14:46,000 --> 00:14:51,000
need later on in this class.
If you want to know everything
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00:14:51,000 --> 00:14:56,000
about the secret life of
matrices, then you should take
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00:14:56,000 --> 00:14:59,000
18.06 someday.
OK, what's going to be our
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00:14:59,000 --> 00:15:02,000
motivation for matrices?
Well, in life,
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00:15:02,000 --> 00:15:07,000
a lot of things are related by
linear formulas.
191
00:15:07,000 --> 00:15:10,000
And, even if they are not,
maybe sometimes you can
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00:15:10,000 --> 00:15:12,000
approximate them by linear
formulas.
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00:15:12,000 --> 00:15:30,000
So, often, we have linear
relations between variables --
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00:15:30,000 --> 00:15:47,000
for example, if we do a change
of coordinate systems.
195
00:15:47,000 --> 00:15:52,000
For example,
say that we are in space,
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00:15:52,000 --> 00:15:58,000
and we have a point.
Its coordinates might be,
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00:15:58,000 --> 00:16:02,000
let me call them x1,
x2, x3 in my initial coordinate
198
00:16:02,000 --> 00:16:04,000
system.
But then, maybe I need to
199
00:16:04,000 --> 00:16:07,000
actually switch to different
coordinates to better solve the
200
00:16:07,000 --> 00:16:09,000
problem because they're more
adapted to other things that
201
00:16:09,000 --> 00:16:13,000
we'll do in the problem.
And so I have other coordinates
202
00:16:13,000 --> 00:16:18,000
axes, and in these new
coordinates, P will have
203
00:16:18,000 --> 00:16:22,000
different coordinates -- let me
call them, say,
204
00:16:22,000 --> 00:16:25,000
u1, u2, u3.
And then, the relation between
205
00:16:25,000 --> 00:16:29,000
the old and the new coordinates
is going to be given by linear
206
00:16:29,000 --> 00:16:33,000
formulas -- at least if I choose
the same origin.
207
00:16:33,000 --> 00:16:38,000
Otherwise, there might be
constant terms,
208
00:16:38,000 --> 00:16:50,000
which I will not insist on.
Let me just give an example.
209
00:16:50,000 --> 00:16:58,000
For example,
maybe, let's say u1 could be 2
210
00:16:58,000 --> 00:17:08,000
x1 3 x2 3 x3.
u2 might be 2 x1 4 x2 5 x3.
211
00:17:08,000 --> 00:17:16,000
u3 might be x1 x2 2 x3.
Do not ask me where these
212
00:17:16,000 --> 00:17:18,000
numbers come from.
I just made them up,
213
00:17:18,000 --> 00:17:23,000
that's just an example of what
might happen.
214
00:17:23,000 --> 00:17:30,000
You can put here your favorite
numbers if you want.
215
00:17:30,000 --> 00:17:35,000
Now, in order to express this
kind of linear relation,
216
00:17:35,000 --> 00:17:39,000
we can use matrices.
A matrix is just a table with
217
00:17:39,000 --> 00:17:45,000
numbers in it.
And we can reformulate this in
218
00:17:45,000 --> 00:17:54,000
terms of matrix multiplication
or matrix product.
219
00:17:54,000 --> 00:18:04,000
So, instead of writing this,
I will write that the matrix
220
00:18:04,000 --> 00:18:11,000
|2,3, 3; 2,4,
5; 1,1, 2| times the vector
221
00:18:11,000 --> 00:18:16,000
***amp***lt;x1,
x2, x3> is equal to
222
00:18:16,000 --> 00:18:22,000
***amp***lt;u1,
u2, u3>.
223
00:18:22,000 --> 00:18:26,000
Hopefully you see that there is
the same information content on
224
00:18:26,000 --> 00:18:29,000
both sides.
I just need to explain to you
225
00:18:29,000 --> 00:18:35,000
what this way of multiplying
tables of numbers means.
226
00:18:35,000 --> 00:18:40,000
Well, what it means is really
that we'll have exactly these
227
00:18:40,000 --> 00:18:45,000
same quantities.
Let me just say that more
228
00:18:45,000 --> 00:18:49,000
symbolically:
so maybe this matrix could be
229
00:18:49,000 --> 00:18:56,000
called A, and this we could call
X, and this one we could call U.
230
00:18:56,000 --> 00:19:00,000
Then we'll say A times X equals
U, which is a lot shorter than
231
00:19:00,000 --> 00:19:03,000
that.
Of course, I need to tell you
232
00:19:03,000 --> 00:19:07,000
what A, X, and U are in terms of
their entries for you to get the
233
00:19:07,000 --> 00:19:11,000
formula.
But it's a convenient notation.
234
00:19:11,000 --> 00:19:17,000
So, what does it mean to do a
matrix product?
235
00:19:17,000 --> 00:19:30,000
The entries in the matrix
product are obtained by taking
236
00:19:30,000 --> 00:19:37,000
dot products.
Let's say we are doing the
237
00:19:37,000 --> 00:19:48,000
product AX.
We do a dot products between
238
00:19:48,000 --> 00:20:00,000
the rows of A and the columns of
X.
239
00:20:00,000 --> 00:20:07,000
Here, A is a 3x3 matrix -- that
just means there's three rows
240
00:20:07,000 --> 00:20:14,000
and three columns.
And X is a column vector,
241
00:20:14,000 --> 00:20:20,000
which we can think of as a 3x1
matrix.
242
00:20:20,000 --> 00:20:27,000
It has three rows and only one
column.
243
00:20:27,000 --> 00:20:31,000
Now, what can we do?
Well, I said we are going to do
244
00:20:31,000 --> 00:20:35,000
a dot product between a row of
A: 2,3, 3, and a column of X:
245
00:20:35,000 --> 00:20:38,000
x1, x2, x3.
That dot product will be two
246
00:20:38,000 --> 00:20:43,000
times x1 plus three times x2
plus three times x3.
247
00:20:43,000 --> 00:20:47,000
OK, it's exactly what we want
to set u1 equal to.
248
00:20:47,000 --> 00:20:51,000
Let's do the second one.
I take the second row of A:
249
00:20:51,000 --> 00:20:55,000
2,4, 5, and I do the dot
product with x1,
250
00:20:55,000 --> 00:20:59,000
x2, x3.
I will get two times x1 plus
251
00:20:59,000 --> 00:21:04,000
four times x2 plus five times
x3, which is u2.
252
00:21:04,000 --> 00:21:10,000
And, same thing with the third
one: one times x1 plus one times
253
00:21:10,000 --> 00:21:18,000
x2 plus two times x3.
So that's matrix multiplication.
254
00:21:18,000 --> 00:21:27,000
Let me restate things more
generally.
255
00:21:27,000 --> 00:21:33,000
If I want to find the entries
of a product of two matrices,
256
00:21:33,000 --> 00:21:38,000
A and B -- I'm saying matrices,
but of course they could be
257
00:21:38,000 --> 00:21:41,000
vectors.
Vectors are now a special case
258
00:21:41,000 --> 00:21:44,000
of matrices, just by taking a
matrix of width one.
259
00:21:44,000 --> 00:21:54,000
So, if I have my matrix A,
and I have my matrix B,
260
00:21:54,000 --> 00:22:01,000
then I will get the product,
AB.
261
00:22:01,000 --> 00:22:08,000
Let's say for example -- this
works in any size -- let's say
262
00:22:08,000 --> 00:22:13,000
that A is a 3x4 matrix.
So, it has three rows,
263
00:22:13,000 --> 00:22:15,000
four columns.
And, here, I'm not going to
264
00:22:15,000 --> 00:22:17,000
give you all the values because
I'm not going to compute
265
00:22:17,000 --> 00:22:19,000
everything.
It would take the rest of the
266
00:22:19,000 --> 00:22:23,000
lecture.
And let's say that B is maybe
267
00:22:23,000 --> 00:22:28,000
size 4x2.
So, it has two columns and four
268
00:22:28,000 --> 00:22:30,000
rows.
And, let's say,
269
00:22:30,000 --> 00:22:33,000
for example,
that we have the second column:
270
00:22:33,000 --> 00:22:36,000
0,3, 0,2.
So, in A times B,
271
00:22:36,000 --> 00:22:43,000
the entries should be the dot
products between these rows and
272
00:22:43,000 --> 00:22:46,000
these columns.
Here, we have two columns.
273
00:22:46,000 --> 00:22:49,000
Here, we have three rows.
So, we should get three times
274
00:22:49,000 --> 00:22:55,000
two different possibilities.
And so the answer will have
275
00:22:55,000 --> 00:22:59,000
size 3x2.
We cannot compute most of them,
276
00:22:59,000 --> 00:23:02,000
because I did not give you
numbers, but one of them we can
277
00:23:02,000 --> 00:23:04,000
compute.
We can compute the value that
278
00:23:04,000 --> 00:23:07,000
goes here, namely,
this one in the second column.
279
00:23:07,000 --> 00:23:13,000
So, I select the second column
of B, and I take the first row
280
00:23:13,000 --> 00:23:16,000
of A, and I find:
1 times 0: 0.
281
00:23:16,000 --> 00:23:20,000
2 times 3: 6,
plus 0, plus 8,
282
00:23:20,000 --> 00:23:28,000
should make 14.
So, this entry right here is 14.
283
00:23:28,000 --> 00:23:34,000
In fact, let me tell you about
another way to set it up so that
284
00:23:34,000 --> 00:23:38,000
you can remember more easily
what goes where.
285
00:23:38,000 --> 00:23:43,000
One way that you can set it up
is you can put A here.
286
00:23:43,000 --> 00:23:49,000
You can put B up here,
and then you will get the
287
00:23:49,000 --> 00:23:53,000
answer here.
And, if you want to find what
288
00:23:53,000 --> 00:23:57,000
goes in a given slot here,
then you just look to its left
289
00:23:57,000 --> 00:24:01,000
and you look above it,
and you do the dot product
290
00:24:01,000 --> 00:24:07,000
between these guys.
That's an easy way to remember.
291
00:24:07,000 --> 00:24:09,000
First of all,
it tells you what the size of
292
00:24:09,000 --> 00:24:11,000
the answer will be.
The size will be what fits
293
00:24:11,000 --> 00:24:14,000
nicely in this box:
it should have the same width
294
00:24:14,000 --> 00:24:18,000
as B and the same height as A.
And second, it tells you which
295
00:24:18,000 --> 00:24:22,000
dot product to compute for each
position.
296
00:24:22,000 --> 00:24:27,000
You just look at what's to the
left, and what's above the given
297
00:24:27,000 --> 00:24:29,000
position.
Now, there's a catch.
298
00:24:29,000 --> 00:24:32,000
Can we multiply anything by
anything?
299
00:24:32,000 --> 00:24:35,000
Well, no.
I wouldn't ask the question
300
00:24:35,000 --> 00:24:38,000
otherwise.
But anyway, to be able to do
301
00:24:38,000 --> 00:24:41,000
this dot product,
we need to have the same number
302
00:24:41,000 --> 00:24:45,000
of entries here and here.
Otherwise, we can't say "take
303
00:24:45,000 --> 00:24:46,000
this times that,
plus this times that,
304
00:24:46,000 --> 00:24:50,000
and so on" if we run out of
space on one of them before the
305
00:24:50,000 --> 00:24:57,000
other.
So, the condition -- and it's
306
00:24:57,000 --> 00:25:12,000
important, so let me write it in
red -- is that the width of A
307
00:25:12,000 --> 00:25:22,000
must equal the height of B.
(OK, it's a bit cluttered,
308
00:25:22,000 --> 00:25:28,000
but hopefully you can still see
what I'm writing.)
309
00:25:28,000 --> 00:25:31,000
OK, now we know how to multiply
matrices.
310
00:25:38,000 --> 00:25:41,000
So, what does it mean to
multiply matrices?
311
00:25:41,000 --> 00:25:47,000
Of course, we've seen in this
example that we can use a matrix
312
00:25:47,000 --> 00:25:52,000
to tell us how to transform from
x's to u's.
313
00:25:52,000 --> 00:25:54,000
And, that's an example of
multiplication.
314
00:25:54,000 --> 00:25:58,000
But now, let's see that we have
two matrices like that telling
315
00:25:58,000 --> 00:26:01,000
us how to transform from
something to something else.
316
00:26:01,000 --> 00:26:02,000
What does it mean to multiply
them?
317
00:26:11,000 --> 00:26:25,000
I claim that the product AB
represents doing first the
318
00:26:25,000 --> 00:26:36,000
transformation B,
then transformation A.
319
00:26:36,000 --> 00:26:37,000
That's a slightly
counterintuitive thing,
320
00:26:37,000 --> 00:26:40,000
because we are used to writing
things from left to right.
321
00:26:40,000 --> 00:26:43,000
Unfortunately,
with matrices,
322
00:26:43,000 --> 00:26:48,000
you multiply things from right
to left.
323
00:26:48,000 --> 00:26:51,000
If you think about it,
say you have two functions,
324
00:26:51,000 --> 00:26:55,000
f and g, and you write f(g(x)),
it really means you apply first
325
00:26:55,000 --> 00:26:59,000
g then f.
It works the same way as that.
326
00:26:59,000 --> 00:27:06,000
OK, so why is this?
Well, if I write AB times X
327
00:27:06,000 --> 00:27:12,000
where X is some vector that I
want to transform,
328
00:27:12,000 --> 00:27:16,000
it's the same as A times BX.
This property is called
329
00:27:16,000 --> 00:27:19,000
associativity.
And, it's a good property of
330
00:27:19,000 --> 00:27:23,000
well-behaved products -- not of
cross product,
331
00:27:23,000 --> 00:27:27,000
by the way.
Matrix product is associative.
332
00:27:27,000 --> 00:27:30,000
That means we can actually
think of a product ABX and
333
00:27:30,000 --> 00:27:32,000
multiply them in whichever order
we want.
334
00:27:32,000 --> 00:27:37,000
We can start with BX or we can
start with AB.
335
00:27:37,000 --> 00:27:43,000
So, now, BX means we apply the
transformation B to X.
336
00:27:43,000 --> 00:27:46,000
And then, multiplying by A
means we apply the
337
00:27:46,000 --> 00:27:49,000
transformation A.
So, we first apply B,
338
00:27:49,000 --> 00:27:58,000
then we apply A.
That's the same as applying AB
339
00:27:58,000 --> 00:28:05,000
all at once.
Another thing -- a warning:
340
00:28:05,000 --> 00:28:10,000
AB and BA are not the same
thing at all.
341
00:28:10,000 --> 00:28:13,000
You can probably see that
already from this
342
00:28:13,000 --> 00:28:18,000
interpretation.
It's not the same thing to
343
00:28:18,000 --> 00:28:24,000
convert oranges to bananas and
then to carrots,
344
00:28:24,000 --> 00:28:28,000
or vice versa.
Actually, even worse:
345
00:28:28,000 --> 00:28:31,000
this thing might not even be
well defined.
346
00:28:31,000 --> 00:28:38,000
If the width of A equals the
height of B, we can do this
347
00:28:38,000 --> 00:28:42,000
product.
But it's not clear that the
348
00:28:42,000 --> 00:28:47,000
width of B will equal the height
of A, which is what we would
349
00:28:47,000 --> 00:28:50,000
need for that one.
So, the size condition,
350
00:28:50,000 --> 00:28:53,000
to be able to do the product,
might not make sense -- maybe
351
00:28:53,000 --> 00:28:56,000
one of the products doesn't make
sense.
352
00:28:56,000 --> 00:29:01,000
Even if they both make sense,
they are usually completely
353
00:29:01,000 --> 00:29:07,000
different things.
The next thing I need to tell
354
00:29:07,000 --> 00:29:13,000
you about is something called
the identity matrix.
355
00:29:13,000 --> 00:29:17,000
The identity matrix is the
matrix that does nothing.
356
00:29:17,000 --> 00:29:19,000
What does it mean to do nothing?
I don't mean the matrix is zero.
357
00:29:19,000 --> 00:29:23,000
The matrix zero would take X
and would always give you back
358
00:29:23,000 --> 00:29:26,000
zero.
That's not a very interesting
359
00:29:26,000 --> 00:29:29,000
transformation.
What I mean is the guy that
360
00:29:29,000 --> 00:29:33,000
takes X and gives you X again.
It's called I,
361
00:29:33,000 --> 00:29:38,000
and it has the property that IX
equals X for all X.
362
00:29:38,000 --> 00:29:41,000
So, it's the transformation
from something to itself.
363
00:29:41,000 --> 00:29:44,000
It's the obvious transformation
-- called the identity
364
00:29:44,000 --> 00:29:48,000
transformation.
So, how do we write that as a
365
00:29:48,000 --> 00:29:51,000
matrix?
Well, actually there's an
366
00:29:51,000 --> 00:29:56,000
identity for each size because,
depending on whether X has two
367
00:29:56,000 --> 00:30:01,000
entries or ten entries,
the matrix I needs to have a
368
00:30:01,000 --> 00:30:05,000
different size.
For example,
369
00:30:05,000 --> 00:30:10,000
the identity matrix of size 3x3
has entries one,
370
00:30:10,000 --> 00:30:15,000
one, one on the diagonal,
and zero everywhere else.
371
00:30:15,000 --> 00:30:22,000
OK, let's check.
If we multiply this with a
372
00:30:22,000 --> 00:30:28,000
vector -- start thinking about
it.
373
00:30:28,000 --> 00:30:31,000
What happens when multiply this
with the vector X?
374
00:31:00,000 --> 00:31:11,000
OK, so let's say I multiply the
matrix I with a vector x1,
375
00:31:11,000 --> 00:31:15,000
x2, x3.
What will the first entry be?
376
00:31:15,000 --> 00:31:19,000
It will be the dot product
between ***amp***lt;1,0,0> and
377
00:31:19,000 --> 00:31:23,000
***amp***lt;x1 x2 x3>.
This vector is i hat.
378
00:31:23,000 --> 00:31:27,000
If you do the dot product with
i hat, you will get the first
379
00:31:27,000 --> 00:31:32,000
component -- that will be x1.
One times x1 plus zero, zero.
380
00:31:32,000 --> 00:31:35,000
Similarly here,
if I do the dot product,
381
00:31:35,000 --> 00:31:40,000
I get zero plus x2 plus zero.
I get x2, and here I get x3.
382
00:31:40,000 --> 00:31:44,000
OK, it works.
Same thing if I put here a
383
00:31:44,000 --> 00:31:48,000
matrix: I will get back the same
matrix.
384
00:31:48,000 --> 00:31:58,000
In general, the identity matrix
in size n x n is an n x n matrix
385
00:31:58,000 --> 00:32:07,000
with ones on the diagonal,
and zeroes everywhere else.
386
00:32:07,000 --> 00:32:11,000
You just put 1 at every
diagonal position and 0
387
00:32:11,000 --> 00:32:13,000
elsewhere.
And then, you can see that if
388
00:32:13,000 --> 00:32:15,000
you multiply that by a vector,
you'll get the same vector
389
00:32:15,000 --> 00:32:15,000
back.
390
00:32:29,000 --> 00:32:39,000
OK, let me give you another
example of a matrix.
391
00:32:39,000 --> 00:32:53,000
Let's say that in the plane we
look at the transformation that
392
00:32:53,000 --> 00:33:05,000
does rotation by 90°,
let's say, counterclockwise.
393
00:33:05,000 --> 00:33:11,000
I claim that this is given by
the matrix: |0,1;
394
00:33:11,000 --> 00:33:19,000
- 1,0|.
Let's try to see why that is
395
00:33:19,000 --> 00:33:25,000
the case.
Well, if I do R times i hat --
396
00:33:25,000 --> 00:33:29,000
if I apply that to the first
vector,
397
00:33:29,000 --> 00:33:35,000
i hat: i hat will be
***amp***lt;1,0> so in this
398
00:33:35,000 --> 00:33:39,000
product,
first you will get 0,
399
00:33:39,000 --> 00:33:46,000
and then you will get 1.
You get j hat.
400
00:33:46,000 --> 00:33:53,000
OK, so this thing sends i hat
to j hat.
401
00:33:53,000 --> 00:34:06,000
What about j hat?
Well, you get negative one.
402
00:34:06,000 --> 00:34:10,000
And then you get 0.
So, that's minus i hat.
403
00:34:10,000 --> 00:34:15,000
So, j is sent towards here.
And, in general,
404
00:34:15,000 --> 00:34:19,000
if you apply it to a vector
with components x,y,
405
00:34:19,000 --> 00:34:29,000
then you will get back -y,x,
which is the formula we've seen
406
00:34:29,000 --> 00:34:39,000
for rotating a vector by 90°.
So, it seems to do what we want.
407
00:34:39,000 --> 00:34:47,000
By the way, the columns in this
matrix represent what happens to
408
00:34:47,000 --> 00:34:53,000
each basis vector,
to the vectors i and j.
409
00:34:53,000 --> 00:34:57,000
This guy here is exactly what
we get when we multiply R by i.
410
00:34:57,000 --> 00:35:05,000
And, when we multiply R by j,
we get this guy here.
411
00:35:05,000 --> 00:35:08,000
So, what's interesting about
this matrix?
412
00:35:08,000 --> 00:35:12,000
Well, we can do computations
with matrices in ways that are
413
00:35:12,000 --> 00:35:15,000
easier than writing coordinate
change formulas.
414
00:35:15,000 --> 00:35:19,000
For example,
if you compute R squared,
415
00:35:19,000 --> 00:35:23,000
so if you multiply R with
itself: I'll let you do it as an
416
00:35:23,000 --> 00:35:28,000
exercise,
but you will find that you get
417
00:35:28,000 --> 00:35:33,000
|-1,0;0,-1|.
So, that's minus the identity
418
00:35:33,000 --> 00:35:35,000
matrix.
Why is that?
419
00:35:35,000 --> 00:35:39,000
Well, if I rotate something by
90° and then I rotate by 90°
420
00:35:39,000 --> 00:35:42,000
again, then I will rotate by
180�.
421
00:35:42,000 --> 00:35:46,000
That means I will actually just
go to the opposite point around
422
00:35:46,000 --> 00:35:51,000
the origin.
So, I will take (x,y) to
423
00:35:51,000 --> 00:35:58,000
(-x,-y).
And if I applied R four times,
424
00:35:58,000 --> 00:36:06,000
R^4 would be identity.
OK, questions?
425
00:36:06,000 --> 00:36:11,000
STUDENT QUESTION:
when you said R equals that
426
00:36:11,000 --> 00:36:14,000
matrix, is that the definition
of R?
427
00:36:14,000 --> 00:36:17,000
How did I come up with this R?
Well, secretly,
428
00:36:17,000 --> 00:36:21,000
I worked pretty hard to find
the entries that would tell me
429
00:36:21,000 --> 00:36:25,000
how to rotate something by 90°
counterclockwise.
430
00:36:25,000 --> 00:36:32,000
So, remember:
what we saw last time or in the
431
00:36:32,000 --> 00:36:39,000
first lecture is that,
to rotate a vector by 90°,
432
00:36:39,000 --> 00:36:46,000
we should change (x,
y) to (-y, x).
433
00:36:46,000 --> 00:36:52,000
And now I'm trying to express
this transformation as a matrix.
434
00:36:52,000 --> 00:36:57,000
So, maybe you can call these
guys u and v,
435
00:36:57,000 --> 00:37:02,000
and then you write that u
equals 0x-1y,
436
00:37:02,000 --> 00:37:08,000
and that v equals 1x 0y.
So that's how I would find it.
437
00:37:08,000 --> 00:37:13,000
Here, I just gave it to you
already made,
438
00:37:13,000 --> 00:37:19,000
so you didn't really see how I
found it.
439
00:37:19,000 --> 00:37:30,000
You will see more about
rotations on the problem set.
440
00:37:30,000 --> 00:37:35,000
OK, next I need to tell you how
to invert matrices.
441
00:37:35,000 --> 00:37:39,000
So, what's the point of
matrices?
442
00:37:39,000 --> 00:37:41,000
It's that it gives us a nice
way to think about changes of
443
00:37:41,000 --> 00:37:43,000
variables.
And, in particular,
444
00:37:43,000 --> 00:37:48,000
if we know how to express U in
terms of X, maybe we'd like to
445
00:37:48,000 --> 00:37:51,000
know how to express X in terms
of U.
446
00:37:51,000 --> 00:37:54,000
Well, we can do that,
because we've learned how to
447
00:37:54,000 --> 00:37:58,000
solve linear systems like this.
So in principle,
448
00:37:58,000 --> 00:38:01,000
we could start working,
substituting and so on,
449
00:38:01,000 --> 00:38:06,000
to find formulas for x1,
x2, x3 as functions of u1,
450
00:38:06,000 --> 00:38:09,000
u2, u3.
And the relation will be,
451
00:38:09,000 --> 00:38:11,000
again, a linear relation.
It will, again,
452
00:38:11,000 --> 00:38:14,000
be given by a matrix.
Well, what's that matrix?
453
00:38:14,000 --> 00:38:17,000
It's the inverse
transformation.
454
00:38:17,000 --> 00:38:21,000
It's the inverse of the matrix
A.
455
00:38:21,000 --> 00:38:24,000
So, we need to learn how to
find the inverse of a matrix
456
00:38:24,000 --> 00:38:25,000
directly.
457
00:38:43,000 --> 00:38:48,000
The inverse of A,
by definition,
458
00:38:48,000 --> 00:38:56,000
is a matrix M,
with the property that if I
459
00:38:56,000 --> 00:39:03,000
multiply A by M,
then I get identity.
460
00:39:03,000 --> 00:39:07,000
And, if I multiply M by A,
I also get identity.
461
00:39:07,000 --> 00:39:10,000
The two properties are
equivalent.
462
00:39:10,000 --> 00:39:13,000
That means, if I apply first
the transformation A,
463
00:39:13,000 --> 00:39:16,000
then the transformation M,
actually I undo the
464
00:39:16,000 --> 00:39:18,000
transformation A,
and vice versa.
465
00:39:18,000 --> 00:39:24,000
These two transformations are
the opposite of each other,
466
00:39:24,000 --> 00:39:28,000
or I should say the inverse of
each other.
467
00:39:28,000 --> 00:39:37,000
For this to make sense,
we need A to be a square
468
00:39:37,000 --> 00:39:41,000
matrix.
It must have size n by n.
469
00:39:41,000 --> 00:39:45,000
It can be any size,
but it must have the same
470
00:39:45,000 --> 00:39:50,000
number of rows as columns.
It's a general fact that you
471
00:39:50,000 --> 00:39:55,000
will see more in detail in
linear algebra if you take it.
472
00:39:55,000 --> 00:40:09,000
Let's just admit it.
The matrix M will be denoted by
473
00:40:09,000 --> 00:40:13,000
A inverse.
Then, what is it good for?
474
00:40:13,000 --> 00:40:18,000
Well, for example,
finding the solution to a
475
00:40:18,000 --> 00:40:21,000
linear system.
What's a linear system in our
476
00:40:21,000 --> 00:40:24,000
new language?
It's: a matrix times some
477
00:40:24,000 --> 00:40:28,000
unknown vector,
X, equals some known vector,
478
00:40:28,000 --> 00:40:32,000
B.
How do we solve that?
479
00:40:32,000 --> 00:40:37,000
We just compute:
X equals A inverse B.
480
00:40:37,000 --> 00:40:42,000
Why does that work?
How do I get from here to here?
481
00:40:42,000 --> 00:40:43,000
Let's be careful.
482
00:40:51,000 --> 00:40:54,000
(I'm going to reuse this
matrix, but I'm going to erase
483
00:40:54,000 --> 00:40:57,000
it nonetheless and I'll just
rewrite it).
484
00:41:21,000 --> 00:41:30,000
If AX=B, then let's multiply
both sides by A inverse.
485
00:41:30,000 --> 00:41:35,000
A inverse times AX is A inverse
B.
486
00:41:35,000 --> 00:41:41,000
And then, A inverse times A is
identity, so I get:
487
00:41:41,000 --> 00:41:46,000
X equals A inverse B.
That's how I solved my system
488
00:41:46,000 --> 00:41:48,000
of equations.
So, if you have a calculator
489
00:41:48,000 --> 00:41:51,000
that can invert matrices,
then you can solve linear
490
00:41:51,000 --> 00:41:55,000
systems very quickly.
Now, we should still learn how
491
00:41:55,000 --> 00:41:58,000
to compute these things.
Yes?
492
00:41:58,000 --> 00:42:03,000
[Student Questions:]"How do you
know that A inverse will be on
493
00:42:03,000 --> 00:42:07,000
the left of B and not after it "
Well,
494
00:42:07,000 --> 00:42:10,000
it's exactly this derivation.
So, if you are not sure,
495
00:42:10,000 --> 00:42:13,000
then just reproduce this
calculation.
496
00:42:13,000 --> 00:42:16,000
To get from here to here,
what I did is I multiplied
497
00:42:16,000 --> 00:42:20,000
things on the left by A inverse,
and then this guy simplify.
498
00:42:20,000 --> 00:42:23,000
If I had put A inverse on the
right, I would have AX A
499
00:42:23,000 --> 00:42:27,000
inverse, which might not make
sense, and even if it makes
500
00:42:27,000 --> 00:42:31,000
sense, it doesn't simplify.
So, the basic rule is that you
501
00:42:31,000 --> 00:42:35,000
have to multiply by A inverse on
the left so that it cancels with
502
00:42:35,000 --> 00:42:38,000
this A that's on the left.
STUDENT QUESTION:
503
00:42:38,000 --> 00:42:41,000
"And if you put it on the left
on this side then it will be on
504
00:42:41,000 --> 00:42:43,000
the left with B as well?" That's
correct,
505
00:42:43,000 --> 00:42:46,000
in our usual way of dealing
with matrices,
506
00:42:46,000 --> 00:42:49,000
where the vectors are column
vectors.
507
00:42:49,000 --> 00:42:52,000
It's just something to
remember: if you have a square
508
00:42:52,000 --> 00:42:56,000
matrix times a column vector,
the product that makes sense is
509
00:42:56,000 --> 00:42:58,000
with the matrix on the left,
and the vector on the right.
510
00:42:58,000 --> 00:43:04,000
The other one just doesn't work.
You cannot take X times A if A
511
00:43:04,000 --> 00:43:11,000
is a square matrix and X is a
column vector.
512
00:43:11,000 --> 00:43:16,000
This product AX makes sense.
The other one XA doesn't make
513
00:43:16,000 --> 00:43:19,000
sense.
It's not the right size.
514
00:43:19,000 --> 00:43:23,000
OK.
What we need to do is to learn
515
00:43:23,000 --> 00:43:29,000
how to invert a matrix.
It's a useful thing to know,
516
00:43:29,000 --> 00:43:32,000
first for your general
knowledge, and second because
517
00:43:32,000 --> 00:43:38,000
it's actually useful for things
we'll see later in this class.
518
00:43:38,000 --> 00:43:40,000
In particular,
on the exam,
519
00:43:40,000 --> 00:43:45,000
you will need to know how to
invert a matrix by hand.
520
00:43:45,000 --> 00:43:50,000
This formula is actually good
for small matrices,
521
00:43:50,000 --> 00:43:52,000
3x3,4x4.
It's not good at all if you
522
00:43:52,000 --> 00:43:54,000
have a matrix of size
1,000x1,000.
523
00:43:54,000 --> 00:43:59,000
So, in computer software,
actually for small matrices
524
00:43:59,000 --> 00:44:02,000
they do this,
but for larger matrices,
525
00:44:02,000 --> 00:44:09,000
they use other algorithms.
Let's just see how we do it.
526
00:44:09,000 --> 00:44:13,000
First of all I will give you
the final answer.
527
00:44:13,000 --> 00:44:19,000
And of course I will need to
explain what the answer means.
528
00:44:19,000 --> 00:44:22,000
We will have to compute
something called the adjoint
529
00:44:22,000 --> 00:44:24,000
matrix.
I will tell you how to do that.
530
00:44:24,000 --> 00:44:35,000
And then, we will divide by the
determinant of A.
531
00:44:35,000 --> 00:44:38,000
How do we get to the adjoint
matrix?
532
00:44:38,000 --> 00:44:46,000
Let's go through the steps on a
3x3 example -- the steps are the
533
00:44:46,000 --> 00:44:52,000
same no matter what the size is,
but let's do 3x3.
534
00:44:52,000 --> 00:44:56,000
So, let's say that I'm giving
you the matrix A -- let's say
535
00:44:56,000 --> 00:44:59,000
it's the same as the one that I
erased earlier.
536
00:44:59,000 --> 00:45:08,000
That was the one relating our
X's and our U's.
537
00:45:08,000 --> 00:45:18,000
The first thing I want to do is
find something called the
538
00:45:18,000 --> 00:45:22,000
minors.
What's a minor?
539
00:45:22,000 --> 00:45:24,000
It's a slightly smaller
determinant.
540
00:45:24,000 --> 00:45:28,000
We've already seen them without
calling them that way.
541
00:45:28,000 --> 00:45:32,000
The matrix of minors will have
again the same size.
542
00:45:32,000 --> 00:45:37,000
Let's say we want this entry.
Then, we just delete this row
543
00:45:37,000 --> 00:45:40,000
and this column,
and we are left with a 2x2
544
00:45:40,000 --> 00:45:44,000
determinant.
So, here, we'll put the
545
00:45:44,000 --> 00:45:49,000
determinant 4,5,
1,2, which is 4 times 2:
546
00:45:49,000 --> 00:45:51,000
8 -- minus 5:
3.
547
00:45:51,000 --> 00:45:53,000
Let's do the next one.
So, for this entry,
548
00:45:53,000 --> 00:45:55,000
I'll delete this row and this
column.
549
00:45:55,000 --> 00:46:00,000
I'm left with 2,5, 1,2.
The determinant will be 2 times
550
00:46:00,000 --> 00:46:04,000
2 minus 5, which is negative 1.
Then minus 2,
551
00:46:04,000 --> 00:46:09,000
then I get to the second row,
so I get to this entry.
552
00:46:09,000 --> 00:46:12,000
To find the minor here,
I will delete this row and this
553
00:46:12,000 --> 00:46:15,000
column.
And I'm left with 3,3, 1,2.
554
00:46:15,000 --> 00:46:24,000
3 times 2 minus 3 is 3.
Let me just cheat and give you
555
00:46:24,000 --> 00:46:31,000
the others -- I think I've shown
you that I can do them.
556
00:46:31,000 --> 00:46:34,000
Let's just explain the last one
again.
557
00:46:34,000 --> 00:46:37,000
The last one is 2.
To find the minor here,
558
00:46:37,000 --> 00:46:41,000
I delete this column and this
row, and I take this
559
00:46:41,000 --> 00:46:44,000
determinant: 2 times 4 minus 2
times 3.
560
00:46:44,000 --> 00:46:49,000
So it's the same kind of
manipulation that we've seen
561
00:46:49,000 --> 00:46:53,000
when we've taken determinants
and cross products.
562
00:46:53,000 --> 00:46:59,000
Step two: we go to another
matrix that's called cofactors.
563
00:46:59,000 --> 00:47:03,000
So, the cofactors are pretty
much the same thing as the
564
00:47:03,000 --> 00:47:07,000
minors except the signs are
slightly different.
565
00:47:07,000 --> 00:47:16,000
What we do is that we flip
signs according to a
566
00:47:16,000 --> 00:47:22,000
checkerboard diagram.
You start with a plus in the
567
00:47:22,000 --> 00:47:26,000
upper left corner,
and you alternate pluses and
568
00:47:26,000 --> 00:47:28,000
minuses.
The rule is:
569
00:47:28,000 --> 00:47:33,000
if there is a plus somewhere,
then there's a minus next to it
570
00:47:33,000 --> 00:47:36,000
and below it.
And then, below a minus or to
571
00:47:36,000 --> 00:47:38,000
the right of a minus,
there's a plus.
572
00:47:38,000 --> 00:47:43,000
So that's how it looks in size
3x3.
573
00:47:43,000 --> 00:47:46,000
What do I mean by that?
I don't mean,
574
00:47:46,000 --> 00:47:48,000
make this positive,
make this negative,
575
00:47:48,000 --> 00:47:50,000
and so on.
That's not what I mean.
576
00:47:50,000 --> 00:47:53,000
What I mean is:
if there's a plus,
577
00:47:53,000 --> 00:47:59,000
that means leave it alone -- we
don't do anything to it.
578
00:47:59,000 --> 00:48:05,000
If there's a minus,
that means we flip the sign.
579
00:48:05,000 --> 00:48:17,000
So, here, we'd get:
3, then 1, -2,
580
00:48:17,000 --> 00:48:25,000
-3,1, 1...
3,-4, and 2.
581
00:48:25,000 --> 00:48:29,000
OK, that step is pretty easy.
The only hard step in terms of
582
00:48:29,000 --> 00:48:32,000
calculations is the first one
because you have to compute all
583
00:48:32,000 --> 00:48:33,000
of these 2x2 determinants.
584
00:48:40,000 --> 00:48:44,000
By the way, this minus sign
here is actually related to the
585
00:48:44,000 --> 00:48:47,000
way in which,
when we do a cross product,
586
00:48:47,000 --> 00:48:51,000
we have a minus sign for the
second entry.
587
00:48:51,000 --> 00:49:00,000
OK, we're almost done.
The third step is to transpose.
588
00:49:00,000 --> 00:49:03,000
What does it mean to transpose?
It means: you read the rows of
589
00:49:03,000 --> 00:49:07,000
your matrix and write them as
columns, or vice versa.
590
00:49:07,000 --> 00:49:16,000
So we switch rows and columns.
What do we get?
591
00:49:16,000 --> 00:49:19,000
Well, let's just read the
matrix horizontally and write it
592
00:49:19,000 --> 00:49:24,000
vertically.
We read 3,1, - 2: 3,1, - 2.
593
00:49:24,000 --> 00:49:29,000
Then we read -3 3,1,
1: - 3,1, 1.
594
00:49:29,000 --> 00:49:39,000
Then, 3, - 4,2: 3, - 4,2.
That's pretty easy.
595
00:49:39,000 --> 00:49:44,000
We're almost done.
What we get here is this is the
596
00:49:44,000 --> 00:49:52,000
adjoint matrix.
So, the fourth and last step is
597
00:49:52,000 --> 00:49:58,000
to divide by the determinant of
A.
598
00:49:58,000 --> 00:50:04,000
We have to compute the
determinant -- the determinant
599
00:50:04,000 --> 00:50:08,000
of A, not the determinant of
this guy.
600
00:50:08,000 --> 00:50:16,000
So: 2,3, 3,2, 4,5, 1,1, 2.
I'll let you check my
601
00:50:16,000 --> 00:50:21,000
computation.
I found that it's equal to 3.
602
00:50:21,000 --> 00:50:30,000
So the final answer is that A
inverse is one third of the
603
00:50:30,000 --> 00:50:35,000
matrix we got there:
|3, - 3,3, 1,1,
604
00:50:35,000 --> 00:50:39,000
- 4, - 2,1, 2|.
Now, remember,
605
00:50:39,000 --> 00:50:43,000
A told us how to find the u's
in terms of the x's.
606
00:50:43,000 --> 00:50:47,000
This tells us how to find x-s
in terms of u-s:
607
00:50:47,000 --> 00:50:52,000
if you multiply x1,x2,x3 by
this you get u1,u2,u3.
608
00:50:52,000 --> 00:50:56,000
It also tells you how to solve
a linear system:
609
00:50:56,000 --> 00:51:03,000
A times X equals something.