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Remember last time -- -- we
learned about the cross product
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of vectors in space.
Remember the definition of
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cross product is in terms of
this determinant det| i hat,
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j hat, k hat,
and then the components of A,
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a1, a2, a3,
and then the components of B|
00:00:57.000 --> 00:01:02.000
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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to remember what the actual
formula is.
00:01:08.000 --> 00:01:12.000
The actual formula is obtained
by expanding the determinant.
00:01:12.000 --> 00:01:19.000
So, we actually get the
determinant of a2,
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a3, b2, b3 times i hat,
minus the determinant of a1,
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a3, b1, b3 times j hat plus the
determinant of a1,
00:01:35.000 --> 00:01:42.000
a2, b1, b2, times k hat.
And we also saw a more
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geometric definition of the
cross product.
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We have learned that the length
of the cross product is equal to
00:01:56.000 --> 00:02:04.000
the area of the parallelogram
with sides A and B.
00:02:17.000 --> 00:02:26.000
We have also learned that the
direction of the cross product
00:02:26.000 --> 00:02:37.000
is given by taking the direction
that's perpendicular to A and B.
00:02:37.000 --> 00:02:42.000
If I draw A and B in a plane
(they determine a plane),
00:02:42.000 --> 00:02:48.000
then the cross product should
go in the direction that's
00:02:48.000 --> 00:02:53.000
perpendicular to that plane.
Now, there's two different
00:02:53.000 --> 00:02:56.000
possible directions that are
perpendicular to a plane.
00:02:56.000 --> 00:03:04.000
And, to decide which one it is,
we use the right-hand rule,
00:03:04.000 --> 00:03:07.000
which says if you extend your
right hand in the direction of
00:03:07.000 --> 00:03:10.000
the vector A,
then curve your fingers in the
00:03:10.000 --> 00:03:14.000
direction of B,
then your thumb will go in the
00:03:14.000 --> 00:03:20.000
direction of the cross product.
One thing I didn't quite get to
00:03:20.000 --> 00:03:26.000
say last time is that there are
some funny manipulation rules.
00:03:26.000 --> 00:03:29.000
What are we allowed to do or
not do with cross products?
00:03:29.000 --> 00:03:35.000
So, let me tell you right away
the most surprising one if
00:03:35.000 --> 00:03:41.000
you've never seen it before:
A cross B and B cross A are not
00:03:41.000 --> 00:03:45.000
the same thing.
Why are they not the same thing?
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Well, one way to see it is to
think geometrically.
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The parallelogram still has the
same area, and it's still in the
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same plane.
So, the cross product is still
00:03:54.000 --> 00:03:58.000
perpendicular to the same plane.
But, what happens is that,
00:03:58.000 --> 00:04:01.000
if you try to apply the
right-hand rule but exchange the
00:04:01.000 --> 00:04:04.000
roles of A and B,
then you will either injure
00:04:04.000 --> 00:04:06.000
yourself,
or your thumb will end up
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pointing in the opposite
direction.
00:04:08.000 --> 00:04:12.000
So, in fact,
B cross A and A cross B are
00:04:12.000 --> 00:04:17.000
opposite of each other.
And you can check that in the
00:04:17.000 --> 00:04:19.000
formula because,
for example,
00:04:19.000 --> 00:04:22.000
the i component is a2 b3 minus
a3 b2.
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If you swap the roles of A and
B, you will also have to change
00:04:27.000 --> 00:04:30.000
the signs.
That's a slightly surprising
00:04:30.000 --> 00:04:33.000
thing, but you will see one
easily adjusts to it.
00:04:33.000 --> 00:04:36.000
It just means one must resist
the temptation to write AxB
00:04:36.000 --> 00:04:40.000
equals BxA.
Whenever you do that,
00:04:40.000 --> 00:04:45.000
put a minus sign.
Now, in particular,
00:04:45.000 --> 00:04:53.000
what happens if I do A cross A?
Well, I will get zero.
00:04:53.000 --> 00:04:54.000
And, there's many ways to see
that.
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One is to use the formula.
Also, you can see that the
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parallelogram formed by A and A
is completely flat,
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and it has area zero.
So, we get the zero vector.
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Hopefully you got practice with
cross products,
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and computing them,
in recitation yesterday.
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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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and I want to find the equation
of the plane that contains them.
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So, say I have P1,
P2, P3, three points in space.
00:05:45.000 --> 00:05:48.000
They determine a plane,
at least if they are not
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aligned, and we would like to
find the equation of the plane
00:05:51.000 --> 00:05:56.000
that they determine.
That means, let's say that we
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have a point,
P, in space with coordinates x,
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y, z.
Well, to find the equation of
00:06:07.000 --> 00:06:14.000
the plane -- -- the plane
containing P1,
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P2, and P3,
we need to find a condition on
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the coordinates x,
y, z,
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telling us whether P is in the
plane or not.
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We have several ways of doing
that.
00:06:44.000 --> 00:06:47.000
For example,
one thing we could do.
00:06:47.000 --> 00:06:51.000
Let me just backtrack to
determinants that we saw last
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time.
One way to think about it is to
00:06:56.000 --> 00:07:03.000
consider these vectors,
P1P2, P1P3, and P1P.
00:07:03.000 --> 00:07:07.000
The question of whether they
are all in the same plane is the
00:07:07.000 --> 00:07:12.000
same as asking ourselves whether
the parallelepiped that they
00:07:12.000 --> 00:07:15.000
form is actually completely
flattened.
00:07:15.000 --> 00:07:18.000
So, if I try to form a
parallelepiped with these three
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sides, and P is not in the
plane, then it will have some
00:07:21.000 --> 00:07:24.000
volume.
But, if P is in the plane,
00:07:24.000 --> 00:07:26.000
then it's actually completely
squished.
00:07:26.000 --> 00:07:31.000
So,one possible answer,
one possible way to think of
00:07:31.000 --> 00:07:37.000
the equation of a plane is that
the determinant of these vectors
00:07:37.000 --> 00:07:42.000
should be zero.
Take the determinant of (vector
00:07:42.000 --> 00:07:48.000
P1P,vector P1P2,vector P1P3)
equals 0 (if you do it in a
00:07:48.000 --> 00:07:53.000
different order it doesn't
really matter).
00:07:53.000 --> 00:07:58.000
One possible way to express the
condition that P is in the plane
00:07:58.000 --> 00:08:02.000
is to say that the determinant
of these three vectors has to be
00:08:02.000 --> 00:08:05.000
zero.
And, if I am given coordinates
00:08:05.000 --> 00:08:07.000
for these points -- I'm not
giving you numbers,
00:08:07.000 --> 00:08:10.000
but if I gave you numbers,
then you would be able to plug
00:08:10.000 --> 00:08:14.000
those numbers in.
So, you could compute these two
00:08:14.000 --> 00:08:16.000
vectors P1P2 and P1P3
explicitly.
00:08:16.000 --> 00:08:19.000
But, of course,
P1P would depend on x,
00:08:19.000 --> 00:08:21.000
y, and z.
So, when you compute the
00:08:21.000 --> 00:08:24.000
determinant, you get a formula
that involves x,
00:08:24.000 --> 00:08:26.000
y, and z.
And you'll find that this
00:08:26.000 --> 00:08:29.000
condition on x,
y, z is the equation of a
00:08:29.000 --> 00:08:32.000
plane.
We're going to see more about
00:08:32.000 --> 00:08:36.000
that pretty soon.
Now, let me tell you a slightly
00:08:36.000 --> 00:08:40.000
faster way of doing it.
Actually, it's not much faster,
00:08:40.000 --> 00:08:44.000
It's pretty much the same
calculation, but it's maybe more
00:08:44.000 --> 00:08:50.000
enlightening.
Let me actually show you a nice
00:08:50.000 --> 00:08:56.000
color picture that I prepared
for this.
00:08:56.000 --> 00:09:00.000
One thing that's on this
picture that I haven't drawn
00:09:00.000 --> 00:09:02.000
before is the normal vector to
the plane.
00:09:02.000 --> 00:09:06.000
Why is that?
Well, let's say that we know
00:09:06.000 --> 00:09:09.000
how to find a vector that's
perpendicular to our plane.
00:09:09.000 --> 00:09:13.000
Then, what does it mean for the
point, P, to be in the plane?
00:09:13.000 --> 00:09:19.000
It means that the direction
from P1 to P has to be
00:09:19.000 --> 00:09:29.000
perpendicular to this vector N.
So here's another solution:
00:09:29.000 --> 00:09:43.000
P is in the plane exactly when
the vector P1P is perpendicular
00:09:43.000 --> 00:09:48.000
to N,
where N is some vector that's
00:09:48.000 --> 00:10:05.000
perpendicular to the plane.
N is called a normal vector.
00:10:05.000 --> 00:10:08.000
How do we rephrase this
condition?
00:10:08.000 --> 00:10:13.000
Well, we've learned how to
detect whether two vectors are
00:10:13.000 --> 00:10:18.000
perpendicular to each other
using dot product (that was the
00:10:18.000 --> 00:10:21.000
first lecture).
These two vectors are
00:10:21.000 --> 00:10:25.000
perpendicular exactly when their
dot product is zero.
00:10:25.000 --> 00:10:32.000
So, concretely,
if we have a point P1 given to
00:10:32.000 --> 00:10:34.000
us,
and say we have been able to
00:10:34.000 --> 00:10:37.000
compute the vector N,
then when we actually compute
00:10:37.000 --> 00:10:40.000
what happens,
here we will have the
00:10:40.000 --> 00:10:41.000
coordinates x,
y, z, of a point P,
00:10:41.000 --> 00:10:44.000
and we will get some condition
on x, y, z.
00:10:44.000 --> 00:10:47.000
That will be the equation of a
plane.
00:10:47.000 --> 00:10:50.000
Now, why are these things the
same?
00:10:50.000 --> 00:10:54.000
Well, before I can tell you
that, I should tell you how to
00:10:54.000 --> 00:10:57.000
find a normal vector.
Maybe you are already starting
00:10:57.000 --> 00:11:01.000
to see what the method should
be, because we know how to find
00:11:01.000 --> 00:11:04.000
a vector perpendicular to two
given vectors.
00:11:04.000 --> 00:11:08.000
We know two vectors in that
plane, for example,
00:11:08.000 --> 00:11:11.000
P1P2, and P1P3.
Actually, I could have used
00:11:11.000 --> 00:11:14.000
another permutation of these
points, but, let's use this.
00:11:14.000 --> 00:11:18.000
So, if I want to find a vector
that's perpendicular to both
00:11:18.000 --> 00:11:22.000
P1P2 and P1P3 at the same time,
all I have to do is take their
00:11:22.000 --> 00:11:27.000
cross product.
So, how do we find a vector
00:11:27.000 --> 00:11:32.000
that's perpendicular to the
plane?
00:11:32.000 --> 00:11:46.000
The answer is just the cross
product P1P2 cross P1P3.
00:11:46.000 --> 00:11:49.000
Say you actually took the
points in a different order,
00:11:49.000 --> 00:11:52.000
and you took P1P3 x P1P2.
You would get,
00:11:52.000 --> 00:11:55.000
of course, the opposite vector.
That is fine.
00:11:55.000 --> 00:11:58.000
Any plane actually has
infinitely many normal vectors.
00:11:58.000 --> 00:12:03.000
You can just multiply a normal
vector by any constant,
00:12:03.000 --> 00:12:07.000
you will still get a normal
vector.
00:12:07.000 --> 00:12:12.000
So, that's going to be one of
the main uses of dot product.
00:12:12.000 --> 00:12:16.000
When we know two vectors in a
plane, it lets us find the
00:12:16.000 --> 00:12:21.000
normal vector to the plane,
and that is what we need to
00:12:21.000 --> 00:12:26.000
find the equation.
Now, why is that the same as
00:12:26.000 --> 00:12:33.000
our first answer over there?
Well, the condition that we
00:12:33.000 --> 00:12:39.000
have is that P1P dot N should be
0.
00:12:39.000 --> 00:12:48.000
And we said N is actually P1P2
cross P1P3.
00:12:48.000 --> 00:12:51.000
So, this is what we want to be
zero.
00:12:51.000 --> 00:12:56.000
Now, if you remember,
a long time ago (that was
00:12:56.000 --> 00:13:04.000
Friday) we've introduced this
thing and called it the triple
00:13:04.000 --> 00:13:07.000
product.
And what we've seen is that the
00:13:07.000 --> 00:13:10.000
triple product is the same thing
as the determinant.
00:13:10.000 --> 00:13:13.000
So, in fact,
these two ways of thinking,
00:13:13.000 --> 00:13:17.000
one saying that the box formed
by these three vectors should be
00:13:17.000 --> 00:13:21.000
flat and have volume zero,
and the other one saying that
00:13:21.000 --> 00:13:25.000
we can find a normal vector and
then express the condition that
00:13:25.000 --> 00:13:29.000
a vector is in the plane if it's
perpendicular to the normal
00:13:29.000 --> 00:13:31.000
vector,
are actually giving us the same
00:13:31.000 --> 00:13:32.000
formula in the end.
00:13:41.000 --> 00:13:46.000
OK, any quick questions before
we move on?
00:13:46.000 --> 00:13:50.000
STUDENT QUESTION:
are those two equal only when P
00:13:50.000 --> 00:13:53.000
is in the plane,
or no matter where it is?
00:13:53.000 --> 00:13:57.000
So, these two quantities,
P1P dot the cross product,
00:13:57.000 --> 00:14:02.000
or the determinant of the three
vectors, are always equal to
00:14:02.000 --> 00:14:04.000
each other.
They are always the same.
00:14:04.000 --> 00:14:08.000
And now, if a point is not in
the plane, then their numerical
00:14:08.000 --> 00:14:13.000
value will be nonzero.
If P is in the plane,
00:14:13.000 --> 00:14:26.000
it will be zero.
OK, let's move on and talk a
00:14:26.000 --> 00:14:35.000
bit about matrices.
Probably some of you have
00:14:35.000 --> 00:14:38.000
learnt about matrices a little
bit in high school,
00:14:38.000 --> 00:14:42.000
but certainly not all of you.
So let me just introduce you to
00:14:42.000 --> 00:14:46.000
a little bit about matrices --
just enough for what we will
00:14:46.000 --> 00:14:51.000
need later on in this class.
If you want to know everything
00:14:51.000 --> 00:14:56.000
about the secret life of
matrices, then you should take
00:14:56.000 --> 00:14:59.000
18.06 someday.
OK, what's going to be our
00:14:59.000 --> 00:15:02.000
motivation for matrices?
Well, in life,
00:15:02.000 --> 00:15:07.000
a lot of things are related by
linear formulas.
00:15:07.000 --> 00:15:10.000
And, even if they are not,
maybe sometimes you can
00:15:10.000 --> 00:15:12.000
approximate them by linear
formulas.
00:15:12.000 --> 00:15:30.000
So, often, we have linear
relations between variables --
00:15:30.000 --> 00:15:47.000
for example, if we do a change
of coordinate systems.
00:15:47.000 --> 00:15:52.000
For example,
say that we are in space,
00:15:52.000 --> 00:15:58.000
and we have a point.
Its coordinates might be,
00:15:58.000 --> 00:16:02.000
let me call them x1,
x2, x3 in my initial coordinate
00:16:02.000 --> 00:16:04.000
system.
But then, maybe I need to
00:16:04.000 --> 00:16:07.000
actually switch to different
coordinates to better solve the
00:16:07.000 --> 00:16:09.000
problem because they're more
adapted to other things that
00:16:09.000 --> 00:16:13.000
we'll do in the problem.
And so I have other coordinates
00:16:13.000 --> 00:16:18.000
axes, and in these new
coordinates, P will have
00:16:18.000 --> 00:16:22.000
different coordinates -- let me
call them, say,
00:16:22.000 --> 00:16:25.000
u1, u2, u3.
And then, the relation between
00:16:25.000 --> 00:16:29.000
the old and the new coordinates
is going to be given by linear
00:16:29.000 --> 00:16:33.000
formulas -- at least if I choose
the same origin.
00:16:33.000 --> 00:16:38.000
Otherwise, there might be
constant terms,
00:16:38.000 --> 00:16:50.000
which I will not insist on.
Let me just give an example.
00:16:50.000 --> 00:16:58.000
For example,
maybe, let's say u1 could be 2
00:16:58.000 --> 00:17:08.000
x1 3 x2 3 x3.
u2 might be 2 x1 4 x2 5 x3.
00:17:08.000 --> 00:17:16.000
u3 might be x1 x2 2 x3.
Do not ask me where these
00:17:16.000 --> 00:17:18.000
numbers come from.
I just made them up,
00:17:18.000 --> 00:17:23.000
that's just an example of what
might happen.
00:17:23.000 --> 00:17:30.000
You can put here your favorite
numbers if you want.
00:17:30.000 --> 00:17:35.000
Now, in order to express this
kind of linear relation,
00:17:35.000 --> 00:17:39.000
we can use matrices.
A matrix is just a table with
00:17:39.000 --> 00:17:45.000
numbers in it.
And we can reformulate this in
00:17:45.000 --> 00:17:54.000
terms of matrix multiplication
or matrix product.
00:17:54.000 --> 00:18:04.000
So, instead of writing this,
I will write that the matrix
00:18:04.000 --> 00:18:11.000
|2,3, 3; 2,4,
5; 1,1, 2| times the vector
00:18:11.000 --> 00:18:16.000
***amp***lt;x1,
x2, x3> is equal to
00:18:16.000 --> 00:18:22.000
***amp***lt;u1,
u2, u3>.
00:18:22.000 --> 00:18:26.000
Hopefully you see that there is
the same information content on
00:18:26.000 --> 00:18:29.000
both sides.
I just need to explain to you
00:18:29.000 --> 00:18:35.000
what this way of multiplying
tables of numbers means.
00:18:35.000 --> 00:18:40.000
Well, what it means is really
that we'll have exactly these
00:18:40.000 --> 00:18:45.000
same quantities.
Let me just say that more
00:18:45.000 --> 00:18:49.000
symbolically:
so maybe this matrix could be
00:18:49.000 --> 00:18:56.000
called A, and this we could call
X, and this one we could call U.
00:18:56.000 --> 00:19:00.000
Then we'll say A times X equals
U, which is a lot shorter than
00:19:00.000 --> 00:19:03.000
that.
Of course, I need to tell you
00:19:03.000 --> 00:19:07.000
what A, X, and U are in terms of
their entries for you to get the
00:19:07.000 --> 00:19:11.000
formula.
But it's a convenient notation.
00:19:11.000 --> 00:19:17.000
So, what does it mean to do a
matrix product?
00:19:17.000 --> 00:19:30.000
The entries in the matrix
product are obtained by taking
00:19:30.000 --> 00:19:37.000
dot products.
Let's say we are doing the
00:19:37.000 --> 00:19:48.000
product AX.
We do a dot products between
00:19:48.000 --> 00:20:00.000
the rows of A and the columns of
X.
00:20:00.000 --> 00:20:07.000
Here, A is a 3x3 matrix -- that
just means there's three rows
00:20:07.000 --> 00:20:14.000
and three columns.
And X is a column vector,
00:20:14.000 --> 00:20:20.000
which we can think of as a 3x1
matrix.
00:20:20.000 --> 00:20:27.000
It has three rows and only one
column.
00:20:27.000 --> 00:20:31.000
Now, what can we do?
Well, I said we are going to do
00:20:31.000 --> 00:20:35.000
a dot product between a row of
A: 2,3, 3, and a column of X:
00:20:35.000 --> 00:20:38.000
x1, x2, x3.
That dot product will be two
00:20:38.000 --> 00:20:43.000
times x1 plus three times x2
plus three times x3.
00:20:43.000 --> 00:20:47.000
OK, it's exactly what we want
to set u1 equal to.
00:20:47.000 --> 00:20:51.000
Let's do the second one.
I take the second row of A:
00:20:51.000 --> 00:20:55.000
2,4, 5, and I do the dot
product with x1,
00:20:55.000 --> 00:20:59.000
x2, x3.
I will get two times x1 plus
00:20:59.000 --> 00:21:04.000
four times x2 plus five times
x3, which is u2.
00:21:04.000 --> 00:21:10.000
And, same thing with the third
one: one times x1 plus one times
00:21:10.000 --> 00:21:18.000
x2 plus two times x3.
So that's matrix multiplication.
00:21:18.000 --> 00:21:27.000
Let me restate things more
generally.
00:21:27.000 --> 00:21:33.000
If I want to find the entries
of a product of two matrices,
00:21:33.000 --> 00:21:38.000
A and B -- I'm saying matrices,
but of course they could be
00:21:38.000 --> 00:21:41.000
vectors.
Vectors are now a special case
00:21:41.000 --> 00:21:44.000
of matrices, just by taking a
matrix of width one.
00:21:44.000 --> 00:21:54.000
So, if I have my matrix A,
and I have my matrix B,
00:21:54.000 --> 00:22:01.000
then I will get the product,
AB.
00:22:01.000 --> 00:22:08.000
Let's say for example -- this
works in any size -- let's say
00:22:08.000 --> 00:22:13.000
that A is a 3x4 matrix.
So, it has three rows,
00:22:13.000 --> 00:22:15.000
four columns.
And, here, I'm not going to
00:22:15.000 --> 00:22:17.000
give you all the values because
I'm not going to compute
00:22:17.000 --> 00:22:19.000
everything.
It would take the rest of the
00:22:19.000 --> 00:22:23.000
lecture.
And let's say that B is maybe
00:22:23.000 --> 00:22:28.000
size 4x2.
So, it has two columns and four
00:22:28.000 --> 00:22:30.000
rows.
And, let's say,
00:22:30.000 --> 00:22:33.000
for example,
that we have the second column:
00:22:33.000 --> 00:22:36.000
0,3, 0,2.
So, in A times B,
00:22:36.000 --> 00:22:43.000
the entries should be the dot
products between these rows and
00:22:43.000 --> 00:22:46.000
these columns.
Here, we have two columns.
00:22:46.000 --> 00:22:49.000
Here, we have three rows.
So, we should get three times
00:22:49.000 --> 00:22:55.000
two different possibilities.
And so the answer will have
00:22:55.000 --> 00:22:59.000
size 3x2.
We cannot compute most of them,
00:22:59.000 --> 00:23:02.000
because I did not give you
numbers, but one of them we can
00:23:02.000 --> 00:23:04.000
compute.
We can compute the value that
00:23:04.000 --> 00:23:07.000
goes here, namely,
this one in the second column.
00:23:07.000 --> 00:23:13.000
So, I select the second column
of B, and I take the first row
00:23:13.000 --> 00:23:16.000
of A, and I find:
1 times 0: 0.
00:23:16.000 --> 00:23:20.000
2 times 3: 6,
plus 0, plus 8,
00:23:20.000 --> 00:23:28.000
should make 14.
So, this entry right here is 14.
00:23:28.000 --> 00:23:34.000
In fact, let me tell you about
another way to set it up so that
00:23:34.000 --> 00:23:38.000
you can remember more easily
what goes where.
00:23:38.000 --> 00:23:43.000
One way that you can set it up
is you can put A here.
00:23:43.000 --> 00:23:49.000
You can put B up here,
and then you will get the
00:23:49.000 --> 00:23:53.000
answer here.
And, if you want to find what
00:23:53.000 --> 00:23:57.000
goes in a given slot here,
then you just look to its left
00:23:57.000 --> 00:24:01.000
and you look above it,
and you do the dot product
00:24:01.000 --> 00:24:07.000
between these guys.
That's an easy way to remember.
00:24:07.000 --> 00:24:09.000
First of all,
it tells you what the size of
00:24:09.000 --> 00:24:11.000
the answer will be.
The size will be what fits
00:24:11.000 --> 00:24:14.000
nicely in this box:
it should have the same width
00:24:14.000 --> 00:24:18.000
as B and the same height as A.
And second, it tells you which
00:24:18.000 --> 00:24:22.000
dot product to compute for each
position.
00:24:22.000 --> 00:24:27.000
You just look at what's to the
left, and what's above the given
00:24:27.000 --> 00:24:29.000
position.
Now, there's a catch.
00:24:29.000 --> 00:24:32.000
Can we multiply anything by
anything?
00:24:32.000 --> 00:24:35.000
Well, no.
I wouldn't ask the question
00:24:35.000 --> 00:24:38.000
otherwise.
But anyway, to be able to do
00:24:38.000 --> 00:24:41.000
this dot product,
we need to have the same number
00:24:41.000 --> 00:24:45.000
of entries here and here.
Otherwise, we can't say "take
00:24:45.000 --> 00:24:46.000
this times that,
plus this times that,
00:24:46.000 --> 00:24:50.000
and so on" if we run out of
space on one of them before the
00:24:50.000 --> 00:24:57.000
other.
So, the condition -- and it's
00:24:57.000 --> 00:25:12.000
important, so let me write it in
red -- is that the width of A
00:25:12.000 --> 00:25:22.000
must equal the height of B.
(OK, it's a bit cluttered,
00:25:22.000 --> 00:25:28.000
but hopefully you can still see
what I'm writing.)
00:25:28.000 --> 00:25:31.000
OK, now we know how to multiply
matrices.
00:25:38.000 --> 00:25:41.000
So, what does it mean to
multiply matrices?
00:25:41.000 --> 00:25:47.000
Of course, we've seen in this
example that we can use a matrix
00:25:47.000 --> 00:25:52.000
to tell us how to transform from
x's to u's.
00:25:52.000 --> 00:25:54.000
And, that's an example of
multiplication.
00:25:54.000 --> 00:25:58.000
But now, let's see that we have
two matrices like that telling
00:25:58.000 --> 00:26:01.000
us how to transform from
something to something else.
00:26:01.000 --> 00:26:02.000
What does it mean to multiply
them?
00:26:11.000 --> 00:26:25.000
I claim that the product AB
represents doing first the
00:26:25.000 --> 00:26:36.000
transformation B,
then transformation A.
00:26:36.000 --> 00:26:37.000
That's a slightly
counterintuitive thing,
00:26:37.000 --> 00:26:40.000
because we are used to writing
things from left to right.
00:26:40.000 --> 00:26:43.000
Unfortunately,
with matrices,
00:26:43.000 --> 00:26:48.000
you multiply things from right
to left.
00:26:48.000 --> 00:26:51.000
If you think about it,
say you have two functions,
00:26:51.000 --> 00:26:55.000
f and g, and you write f(g(x)),
it really means you apply first
00:26:55.000 --> 00:26:59.000
g then f.
It works the same way as that.
00:26:59.000 --> 00:27:06.000
OK, so why is this?
Well, if I write AB times X
00:27:06.000 --> 00:27:12.000
where X is some vector that I
want to transform,
00:27:12.000 --> 00:27:16.000
it's the same as A times BX.
This property is called
00:27:16.000 --> 00:27:19.000
associativity.
And, it's a good property of
00:27:19.000 --> 00:27:23.000
well-behaved products -- not of
cross product,
00:27:23.000 --> 00:27:27.000
by the way.
Matrix product is associative.
00:27:27.000 --> 00:27:30.000
That means we can actually
think of a product ABX and
00:27:30.000 --> 00:27:32.000
multiply them in whichever order
we want.
00:27:32.000 --> 00:27:37.000
We can start with BX or we can
start with AB.
00:27:37.000 --> 00:27:43.000
So, now, BX means we apply the
transformation B to X.
00:27:43.000 --> 00:27:46.000
And then, multiplying by A
means we apply the
00:27:46.000 --> 00:27:49.000
transformation A.
So, we first apply B,
00:27:49.000 --> 00:27:58.000
then we apply A.
That's the same as applying AB
00:27:58.000 --> 00:28:05.000
all at once.
Another thing -- a warning:
00:28:05.000 --> 00:28:10.000
AB and BA are not the same
thing at all.
00:28:10.000 --> 00:28:13.000
You can probably see that
already from this
00:28:13.000 --> 00:28:18.000
interpretation.
It's not the same thing to
00:28:18.000 --> 00:28:24.000
convert oranges to bananas and
then to carrots,
00:28:24.000 --> 00:28:28.000
or vice versa.
Actually, even worse:
00:28:28.000 --> 00:28:31.000
this thing might not even be
well defined.
00:28:31.000 --> 00:28:38.000
If the width of A equals the
height of B, we can do this
00:28:38.000 --> 00:28:42.000
product.
But it's not clear that the
00:28:42.000 --> 00:28:47.000
width of B will equal the height
of A, which is what we would
00:28:47.000 --> 00:28:50.000
need for that one.
So, the size condition,
00:28:50.000 --> 00:28:53.000
to be able to do the product,
might not make sense -- maybe
00:28:53.000 --> 00:28:56.000
one of the products doesn't make
sense.
00:28:56.000 --> 00:29:01.000
Even if they both make sense,
they are usually completely
00:29:01.000 --> 00:29:07.000
different things.
The next thing I need to tell
00:29:07.000 --> 00:29:13.000
you about is something called
the identity matrix.
00:29:13.000 --> 00:29:17.000
The identity matrix is the
matrix that does nothing.
00:29:17.000 --> 00:29:19.000
What does it mean to do nothing?
I don't mean the matrix is zero.
00:29:19.000 --> 00:29:23.000
The matrix zero would take X
and would always give you back
00:29:23.000 --> 00:29:26.000
zero.
That's not a very interesting
00:29:26.000 --> 00:29:29.000
transformation.
What I mean is the guy that
00:29:29.000 --> 00:29:33.000
takes X and gives you X again.
It's called I,
00:29:33.000 --> 00:29:38.000
and it has the property that IX
equals X for all X.
00:29:38.000 --> 00:29:41.000
So, it's the transformation
from something to itself.
00:29:41.000 --> 00:29:44.000
It's the obvious transformation
-- called the identity
00:29:44.000 --> 00:29:48.000
transformation.
So, how do we write that as a
00:29:48.000 --> 00:29:51.000
matrix?
Well, actually there's an
00:29:51.000 --> 00:29:56.000
identity for each size because,
depending on whether X has two
00:29:56.000 --> 00:30:01.000
entries or ten entries,
the matrix I needs to have a
00:30:01.000 --> 00:30:05.000
different size.
For example,
00:30:05.000 --> 00:30:10.000
the identity matrix of size 3x3
has entries one,
00:30:10.000 --> 00:30:15.000
one, one on the diagonal,
and zero everywhere else.
00:30:15.000 --> 00:30:22.000
OK, let's check.
If we multiply this with a
00:30:22.000 --> 00:30:28.000
vector -- start thinking about
it.
00:30:28.000 --> 00:30:31.000
What happens when multiply this
with the vector X?
00:31:00.000 --> 00:31:11.000
OK, so let's say I multiply the
matrix I with a vector x1,
00:31:11.000 --> 00:31:15.000
x2, x3.
What will the first entry be?
00:31:15.000 --> 00:31:19.000
It will be the dot product
between ***amp***lt;1,0,0> and
00:31:19.000 --> 00:31:23.000
***amp***lt;x1 x2 x3>.
This vector is i hat.
00:31:23.000 --> 00:31:27.000
If you do the dot product with
i hat, you will get the first
00:31:27.000 --> 00:31:32.000
component -- that will be x1.
One times x1 plus zero, zero.
00:31:32.000 --> 00:31:35.000
Similarly here,
if I do the dot product,
00:31:35.000 --> 00:31:40.000
I get zero plus x2 plus zero.
I get x2, and here I get x3.
00:31:40.000 --> 00:31:44.000
OK, it works.
Same thing if I put here a
00:31:44.000 --> 00:31:48.000
matrix: I will get back the same
matrix.
00:31:48.000 --> 00:31:58.000
In general, the identity matrix
in size n x n is an n x n matrix
00:31:58.000 --> 00:32:07.000
with ones on the diagonal,
and zeroes everywhere else.
00:32:07.000 --> 00:32:11.000
You just put 1 at every
diagonal position and 0
00:32:11.000 --> 00:32:13.000
elsewhere.
And then, you can see that if
00:32:13.000 --> 00:32:15.000
you multiply that by a vector,
you'll get the same vector
00:32:15.000 --> 00:32:15.000
back.
00:32:29.000 --> 00:32:39.000
OK, let me give you another
example of a matrix.
00:32:39.000 --> 00:32:53.000
Let's say that in the plane we
look at the transformation that
00:32:53.000 --> 00:33:05.000
does rotation by 90°,
let's say, counterclockwise.
00:33:05.000 --> 00:33:11.000
I claim that this is given by
the matrix: |0,1;
00:33:11.000 --> 00:33:19.000
- 1,0|.
Let's try to see why that is
00:33:19.000 --> 00:33:25.000
the case.
Well, if I do R times i hat --
00:33:25.000 --> 00:33:29.000
if I apply that to the first
vector,
00:33:29.000 --> 00:33:35.000
i hat: i hat will be
***amp***lt;1,0> so in this
00:33:35.000 --> 00:33:39.000
product,
first you will get 0,
00:33:39.000 --> 00:33:46.000
and then you will get 1.
You get j hat.
00:33:46.000 --> 00:33:53.000
OK, so this thing sends i hat
to j hat.
00:33:53.000 --> 00:34:06.000
What about j hat?
Well, you get negative one.
00:34:06.000 --> 00:34:10.000
And then you get 0.
So, that's minus i hat.
00:34:10.000 --> 00:34:15.000
So, j is sent towards here.
And, in general,
00:34:15.000 --> 00:34:19.000
if you apply it to a vector
with components x,y,
00:34:19.000 --> 00:34:29.000
then you will get back -y,x,
which is the formula we've seen
00:34:29.000 --> 00:34:39.000
for rotating a vector by 90°.
So, it seems to do what we want.
00:34:39.000 --> 00:34:47.000
By the way, the columns in this
matrix represent what happens to
00:34:47.000 --> 00:34:53.000
each basis vector,
to the vectors i and j.
00:34:53.000 --> 00:34:57.000
This guy here is exactly what
we get when we multiply R by i.
00:34:57.000 --> 00:35:05.000
And, when we multiply R by j,
we get this guy here.
00:35:05.000 --> 00:35:08.000
So, what's interesting about
this matrix?
00:35:08.000 --> 00:35:12.000
Well, we can do computations
with matrices in ways that are
00:35:12.000 --> 00:35:15.000
easier than writing coordinate
change formulas.
00:35:15.000 --> 00:35:19.000
For example,
if you compute R squared,
00:35:19.000 --> 00:35:23.000
so if you multiply R with
itself: I'll let you do it as an
00:35:23.000 --> 00:35:28.000
exercise,
but you will find that you get
00:35:28.000 --> 00:35:33.000
|-1,0;0,-1|.
So, that's minus the identity
00:35:33.000 --> 00:35:35.000
matrix.
Why is that?
00:35:35.000 --> 00:35:39.000
Well, if I rotate something by
90° and then I rotate by 90°
00:35:39.000 --> 00:35:42.000
again, then I will rotate by
180�.
00:35:42.000 --> 00:35:46.000
That means I will actually just
go to the opposite point around
00:35:46.000 --> 00:35:51.000
the origin.
So, I will take (x,y) to
00:35:51.000 --> 00:35:58.000
(-x,-y).
And if I applied R four times,
00:35:58.000 --> 00:36:06.000
R^4 would be identity.
OK, questions?
00:36:06.000 --> 00:36:11.000
STUDENT QUESTION:
when you said R equals that
00:36:11.000 --> 00:36:14.000
matrix, is that the definition
of R?
00:36:14.000 --> 00:36:17.000
How did I come up with this R?
Well, secretly,
00:36:17.000 --> 00:36:21.000
I worked pretty hard to find
the entries that would tell me
00:36:21.000 --> 00:36:25.000
how to rotate something by 90°
counterclockwise.
00:36:25.000 --> 00:36:32.000
So, remember:
what we saw last time or in the
00:36:32.000 --> 00:36:39.000
first lecture is that,
to rotate a vector by 90°,
00:36:39.000 --> 00:36:46.000
we should change (x,
y) to (-y, x).
00:36:46.000 --> 00:36:52.000
And now I'm trying to express
this transformation as a matrix.
00:36:52.000 --> 00:36:57.000
So, maybe you can call these
guys u and v,
00:36:57.000 --> 00:37:02.000
and then you write that u
equals 0x-1y,
00:37:02.000 --> 00:37:08.000
and that v equals 1x 0y.
So that's how I would find it.
00:37:08.000 --> 00:37:13.000
Here, I just gave it to you
already made,
00:37:13.000 --> 00:37:19.000
so you didn't really see how I
found it.
00:37:19.000 --> 00:37:30.000
You will see more about
rotations on the problem set.
00:37:30.000 --> 00:37:35.000
OK, next I need to tell you how
to invert matrices.
00:37:35.000 --> 00:37:39.000
So, what's the point of
matrices?
00:37:39.000 --> 00:37:41.000
It's that it gives us a nice
way to think about changes of
00:37:41.000 --> 00:37:43.000
variables.
And, in particular,
00:37:43.000 --> 00:37:48.000
if we know how to express U in
terms of X, maybe we'd like to
00:37:48.000 --> 00:37:51.000
know how to express X in terms
of U.
00:37:51.000 --> 00:37:54.000
Well, we can do that,
because we've learned how to
00:37:54.000 --> 00:37:58.000
solve linear systems like this.
So in principle,
00:37:58.000 --> 00:38:01.000
we could start working,
substituting and so on,
00:38:01.000 --> 00:38:06.000
to find formulas for x1,
x2, x3 as functions of u1,
00:38:06.000 --> 00:38:09.000
u2, u3.
And the relation will be,
00:38:09.000 --> 00:38:11.000
again, a linear relation.
It will, again,
00:38:11.000 --> 00:38:14.000
be given by a matrix.
Well, what's that matrix?
00:38:14.000 --> 00:38:17.000
It's the inverse
transformation.
00:38:17.000 --> 00:38:21.000
It's the inverse of the matrix
A.
00:38:21.000 --> 00:38:24.000
So, we need to learn how to
find the inverse of a matrix
00:38:24.000 --> 00:38:25.000
directly.
00:38:43.000 --> 00:38:48.000
The inverse of A,
by definition,
00:38:48.000 --> 00:38:56.000
is a matrix M,
with the property that if I
00:38:56.000 --> 00:39:03.000
multiply A by M,
then I get identity.
00:39:03.000 --> 00:39:07.000
And, if I multiply M by A,
I also get identity.
00:39:07.000 --> 00:39:10.000
The two properties are
equivalent.
00:39:10.000 --> 00:39:13.000
That means, if I apply first
the transformation A,
00:39:13.000 --> 00:39:16.000
then the transformation M,
actually I undo the
00:39:16.000 --> 00:39:18.000
transformation A,
and vice versa.
00:39:18.000 --> 00:39:24.000
These two transformations are
the opposite of each other,
00:39:24.000 --> 00:39:28.000
or I should say the inverse of
each other.
00:39:28.000 --> 00:39:37.000
For this to make sense,
we need A to be a square
00:39:37.000 --> 00:39:41.000
matrix.
It must have size n by n.
00:39:41.000 --> 00:39:45.000
It can be any size,
but it must have the same
00:39:45.000 --> 00:39:50.000
number of rows as columns.
It's a general fact that you
00:39:50.000 --> 00:39:55.000
will see more in detail in
linear algebra if you take it.
00:39:55.000 --> 00:40:09.000
Let's just admit it.
The matrix M will be denoted by
00:40:09.000 --> 00:40:13.000
A inverse.
Then, what is it good for?
00:40:13.000 --> 00:40:18.000
Well, for example,
finding the solution to a
00:40:18.000 --> 00:40:21.000
linear system.
What's a linear system in our
00:40:21.000 --> 00:40:24.000
new language?
It's: a matrix times some
00:40:24.000 --> 00:40:28.000
unknown vector,
X, equals some known vector,
00:40:28.000 --> 00:40:32.000
B.
How do we solve that?
00:40:32.000 --> 00:40:37.000
We just compute:
X equals A inverse B.
00:40:37.000 --> 00:40:42.000
Why does that work?
How do I get from here to here?
00:40:42.000 --> 00:40:43.000
Let's be careful.
00:40:51.000 --> 00:40:54.000
(I'm going to reuse this
matrix, but I'm going to erase
00:40:54.000 --> 00:40:57.000
it nonetheless and I'll just
rewrite it).
00:41:21.000 --> 00:41:30.000
If AX=B, then let's multiply
both sides by A inverse.
00:41:30.000 --> 00:41:35.000
A inverse times AX is A inverse
B.
00:41:35.000 --> 00:41:41.000
And then, A inverse times A is
identity, so I get:
00:41:41.000 --> 00:41:46.000
X equals A inverse B.
That's how I solved my system
00:41:46.000 --> 00:41:48.000
of equations.
So, if you have a calculator
00:41:48.000 --> 00:41:51.000
that can invert matrices,
then you can solve linear
00:41:51.000 --> 00:41:55.000
systems very quickly.
Now, we should still learn how
00:41:55.000 --> 00:41:58.000
to compute these things.
Yes?
00:41:58.000 --> 00:42:03.000
[Student Questions:]"How do you
know that A inverse will be on
00:42:03.000 --> 00:42:07.000
the left of B and not after it "
Well,
00:42:07.000 --> 00:42:10.000
it's exactly this derivation.
So, if you are not sure,
00:42:10.000 --> 00:42:13.000
then just reproduce this
calculation.
00:42:13.000 --> 00:42:16.000
To get from here to here,
what I did is I multiplied
00:42:16.000 --> 00:42:20.000
things on the left by A inverse,
and then this guy simplify.
00:42:20.000 --> 00:42:23.000
If I had put A inverse on the
right, I would have AX A
00:42:23.000 --> 00:42:27.000
inverse, which might not make
sense, and even if it makes
00:42:27.000 --> 00:42:31.000
sense, it doesn't simplify.
So, the basic rule is that you
00:42:31.000 --> 00:42:35.000
have to multiply by A inverse on
the left so that it cancels with
00:42:35.000 --> 00:42:38.000
this A that's on the left.
STUDENT QUESTION:
00:42:38.000 --> 00:42:41.000
"And if you put it on the left
on this side then it will be on
00:42:41.000 --> 00:42:43.000
the left with B as well?" That's
correct,
00:42:43.000 --> 00:42:46.000
in our usual way of dealing
with matrices,
00:42:46.000 --> 00:42:49.000
where the vectors are column
vectors.
00:42:49.000 --> 00:42:52.000
It's just something to
remember: if you have a square
00:42:52.000 --> 00:42:56.000
matrix times a column vector,
the product that makes sense is
00:42:56.000 --> 00:42:58.000
with the matrix on the left,
and the vector on the right.
00:42:58.000 --> 00:43:04.000
The other one just doesn't work.
You cannot take X times A if A
00:43:04.000 --> 00:43:11.000
is a square matrix and X is a
column vector.
00:43:11.000 --> 00:43:16.000
This product AX makes sense.
The other one XA doesn't make
00:43:16.000 --> 00:43:19.000
sense.
It's not the right size.
00:43:19.000 --> 00:43:23.000
OK.
What we need to do is to learn
00:43:23.000 --> 00:43:29.000
how to invert a matrix.
It's a useful thing to know,
00:43:29.000 --> 00:43:32.000
first for your general
knowledge, and second because
00:43:32.000 --> 00:43:38.000
it's actually useful for things
we'll see later in this class.
00:43:38.000 --> 00:43:40.000
In particular,
on the exam,
00:43:40.000 --> 00:43:45.000
you will need to know how to
invert a matrix by hand.
00:43:45.000 --> 00:43:50.000
This formula is actually good
for small matrices,
00:43:50.000 --> 00:43:52.000
3x3,4x4.
It's not good at all if you
00:43:52.000 --> 00:43:54.000
have a matrix of size
1,000x1,000.
00:43:54.000 --> 00:43:59.000
So, in computer software,
actually for small matrices
00:43:59.000 --> 00:44:02.000
they do this,
but for larger matrices,
00:44:02.000 --> 00:44:09.000
they use other algorithms.
Let's just see how we do it.
00:44:09.000 --> 00:44:13.000
First of all I will give you
the final answer.
00:44:13.000 --> 00:44:19.000
And of course I will need to
explain what the answer means.
00:44:19.000 --> 00:44:22.000
We will have to compute
something called the adjoint
00:44:22.000 --> 00:44:24.000
matrix.
I will tell you how to do that.
00:44:24.000 --> 00:44:35.000
And then, we will divide by the
determinant of A.
00:44:35.000 --> 00:44:38.000
How do we get to the adjoint
matrix?
00:44:38.000 --> 00:44:46.000
Let's go through the steps on a
3x3 example -- the steps are the
00:44:46.000 --> 00:44:52.000
same no matter what the size is,
but let's do 3x3.
00:44:52.000 --> 00:44:56.000
So, let's say that I'm giving
you the matrix A -- let's say
00:44:56.000 --> 00:44:59.000
it's the same as the one that I
erased earlier.
00:44:59.000 --> 00:45:08.000
That was the one relating our
X's and our U's.
00:45:08.000 --> 00:45:18.000
The first thing I want to do is
find something called the
00:45:18.000 --> 00:45:22.000
minors.
What's a minor?
00:45:22.000 --> 00:45:24.000
It's a slightly smaller
determinant.
00:45:24.000 --> 00:45:28.000
We've already seen them without
calling them that way.
00:45:28.000 --> 00:45:32.000
The matrix of minors will have
again the same size.
00:45:32.000 --> 00:45:37.000
Let's say we want this entry.
Then, we just delete this row
00:45:37.000 --> 00:45:40.000
and this column,
and we are left with a 2x2
00:45:40.000 --> 00:45:44.000
determinant.
So, here, we'll put the
00:45:44.000 --> 00:45:49.000
determinant 4,5,
1,2, which is 4 times 2:
00:45:49.000 --> 00:45:51.000
8 -- minus 5:
3.
00:45:51.000 --> 00:45:53.000
Let's do the next one.
So, for this entry,
00:45:53.000 --> 00:45:55.000
I'll delete this row and this
column.
00:45:55.000 --> 00:46:00.000
I'm left with 2,5, 1,2.
The determinant will be 2 times
00:46:00.000 --> 00:46:04.000
2 minus 5, which is negative 1.
Then minus 2,
00:46:04.000 --> 00:46:09.000
then I get to the second row,
so I get to this entry.
00:46:09.000 --> 00:46:12.000
To find the minor here,
I will delete this row and this
00:46:12.000 --> 00:46:15.000
column.
And I'm left with 3,3, 1,2.
00:46:15.000 --> 00:46:24.000
3 times 2 minus 3 is 3.
Let me just cheat and give you
00:46:24.000 --> 00:46:31.000
the others -- I think I've shown
you that I can do them.
00:46:31.000 --> 00:46:34.000
Let's just explain the last one
again.
00:46:34.000 --> 00:46:37.000
The last one is 2.
To find the minor here,
00:46:37.000 --> 00:46:41.000
I delete this column and this
row, and I take this
00:46:41.000 --> 00:46:44.000
determinant: 2 times 4 minus 2
times 3.
00:46:44.000 --> 00:46:49.000
So it's the same kind of
manipulation that we've seen
00:46:49.000 --> 00:46:53.000
when we've taken determinants
and cross products.
00:46:53.000 --> 00:46:59.000
Step two: we go to another
matrix that's called cofactors.
00:46:59.000 --> 00:47:03.000
So, the cofactors are pretty
much the same thing as the
00:47:03.000 --> 00:47:07.000
minors except the signs are
slightly different.
00:47:07.000 --> 00:47:16.000
What we do is that we flip
signs according to a
00:47:16.000 --> 00:47:22.000
checkerboard diagram.
You start with a plus in the
00:47:22.000 --> 00:47:26.000
upper left corner,
and you alternate pluses and
00:47:26.000 --> 00:47:28.000
minuses.
The rule is:
00:47:28.000 --> 00:47:33.000
if there is a plus somewhere,
then there's a minus next to it
00:47:33.000 --> 00:47:36.000
and below it.
And then, below a minus or to
00:47:36.000 --> 00:47:38.000
the right of a minus,
there's a plus.
00:47:38.000 --> 00:47:43.000
So that's how it looks in size
3x3.
00:47:43.000 --> 00:47:46.000
What do I mean by that?
I don't mean,
00:47:46.000 --> 00:47:48.000
make this positive,
make this negative,
00:47:48.000 --> 00:47:50.000
and so on.
That's not what I mean.
00:47:50.000 --> 00:47:53.000
What I mean is:
if there's a plus,
00:47:53.000 --> 00:47:59.000
that means leave it alone -- we
don't do anything to it.
00:47:59.000 --> 00:48:05.000
If there's a minus,
that means we flip the sign.
00:48:05.000 --> 00:48:17.000
So, here, we'd get:
3, then 1, -2,
00:48:17.000 --> 00:48:25.000
-3,1, 1...
3,-4, and 2.
00:48:25.000 --> 00:48:29.000
OK, that step is pretty easy.
The only hard step in terms of
00:48:29.000 --> 00:48:32.000
calculations is the first one
because you have to compute all
00:48:32.000 --> 00:48:33.000
of these 2x2 determinants.
00:48:40.000 --> 00:48:44.000
By the way, this minus sign
here is actually related to the
00:48:44.000 --> 00:48:47.000
way in which,
when we do a cross product,
00:48:47.000 --> 00:48:51.000
we have a minus sign for the
second entry.
00:48:51.000 --> 00:49:00.000
OK, we're almost done.
The third step is to transpose.
00:49:00.000 --> 00:49:03.000
What does it mean to transpose?
It means: you read the rows of
00:49:03.000 --> 00:49:07.000
your matrix and write them as
columns, or vice versa.
00:49:07.000 --> 00:49:16.000
So we switch rows and columns.
What do we get?
00:49:16.000 --> 00:49:19.000
Well, let's just read the
matrix horizontally and write it
00:49:19.000 --> 00:49:24.000
vertically.
We read 3,1, - 2: 3,1, - 2.
00:49:24.000 --> 00:49:29.000
Then we read -3 3,1,
1: - 3,1, 1.
00:49:29.000 --> 00:49:39.000
Then, 3, - 4,2: 3, - 4,2.
That's pretty easy.
00:49:39.000 --> 00:49:44.000
We're almost done.
What we get here is this is the
00:49:44.000 --> 00:49:52.000
adjoint matrix.
So, the fourth and last step is
00:49:52.000 --> 00:49:58.000
to divide by the determinant of
A.
00:49:58.000 --> 00:50:04.000
We have to compute the
determinant -- the determinant
00:50:04.000 --> 00:50:08.000
of A, not the determinant of
this guy.
00:50:08.000 --> 00:50:16.000
So: 2,3, 3,2, 4,5, 1,1, 2.
I'll let you check my
00:50:16.000 --> 00:50:21.000
computation.
I found that it's equal to 3.
00:50:21.000 --> 00:50:30.000
So the final answer is that A
inverse is one third of the
00:50:30.000 --> 00:50:35.000
matrix we got there:
|3, - 3,3, 1,1,
00:50:35.000 --> 00:50:39.000
- 4, - 2,1, 2|.
Now, remember,
00:50:39.000 --> 00:50:43.000
A told us how to find the u's
in terms of the x's.
00:50:43.000 --> 00:50:47.000
This tells us how to find x-s
in terms of u-s:
00:50:47.000 --> 00:50:52.000
if you multiply x1,x2,x3 by
this you get u1,u2,u3.
00:50:52.000 --> 00:50:56.000
It also tells you how to solve
a linear system:
00:50:56.000 --> 00:51:03.000
A times X equals something.