This video also is going to deal with linear transformation, but we're going to stick with
a linear transformation that operates on vectors in R2 and returns a vector in R2. So these
transformations will be easy to picture as points in a two-dimensional plane being moved
around. Specifically, the linear transformation that we're going to look at in this video
is the transformation T that takes a vector from R2 and returns to R2.
And what it does is it rotates points about the origin in a clockwise manner and it rotates
them by pi over 6 radians. After rotating by pi over 6 radians clockwise it then reflects
that point through the line described by the equation x1 equals a negative x2. So it's
a single linear transformation. But that single linear transformation can be decomposed into
two steps, a rotation followed by a reflection. What we're going to do is, we're going to
find the matrix representation of this linear transformation. And then we're done doing
that, we're actually going to compute what is the linear transformation of the point
1, 0. What happens to this 1, 0, that's a point
on the x1-axis, as it gets rotated and reflected? So we'll get to visualize what this linear
transformation does at the end. So the first component of this linear transformation is
a rotation. So a rotation in a clockwise direction about
the origin can be described by this matrix, cosine theta, sine theta, minus sine theta,
cosine theta, where theta is the amount of the rotation performed in terms of radians.
This is a clockwise rotation. If it was counterclockwise, we would have
a negative sine here and positive sine here. So it's easy to switch this rotation matrix
from a clockwise to a counterclockwise as needed. For our specific case, we're not rotating
by theta radians, we're rotating by theta equals pi over 6.
So if we replace all the thetas with pi over 6 and compute cosine theta and sine theta,
our rotation matrix turns into this matrix right here. So if we were just doing a single
rotation, we'd be done. This would be our matrix representation. But this is just one
part of the full linear transformation in this problem.
The other part that's happening that we haven't talked about yet is the reflection. We're
going to reflect about the line x2 equals a negative x1, or you can think of it as x1
equals a negative x2, same equation. That operation is described by this reflection
matrix here 0, negative 1, 1-- negative 1, 0, I'm sorry.
Let's check. Let's think about why this works. Any time you want to know how a linear transformation
is operating on something, you can just pick an arbitrary element x. In this case, x has
just two components, x1 and x2, and we just operate on this arbitrary vector to see what
happens. So if we start with the vector x1, x2, and
then linearly transform it by this reflection operation, we end up with x1 times the first
column plus x2 times the second column, which turns into a negative x2 and a negative x1.
So you can see what happened. Originally, in our vector, we had x1 here. x1 has changed
to the second coordinate and been negated. Same thing with x2. Originally, we had x2
here in the second coordinate. Now, x2 is in the first coordinate and it's been negated.
Well, that's exactly what our equation says should happen-- x1 should turn into a negative
x2, x2 should turn into a negative x1. So this matrix is doing exactly what we want
it to do in terms of reflecting it about the line that we're supposed to reflect it about.
So that makes sense. So put these together now. Our final matrix
representation of this linear transformation is just the product of these two individual
operations. The first thing that happens is we rotate. So we have to write that on the
far right. And then the next thing that happens is the reflection.
So the order of this is very important. Since we're going to be multiplying by a vector
here, the first thing right here needs to be the first operation, A rot, followed by
A ref. So you kind of think of this stack as the last operation, the next to last operation,
all the way down the line in that order. So let's go ahead and compute what this is.
So this is just these two matrices multiplied together. And if we do that, we end up with
this final matrix for A. So this is the matrix representation of the linear transformation
that we were trying to find. Let's go and do the second part of the problem.
Now let's think about what happens when this linear transformation operates on the point
of 1, 0. Well, we know what that means. That means, take the vector 1, 0 and multiply it
on the left by the matrix A. So that's this quantity right here. If we
actually do that multiplication, when we multiply A by 1, 0 that really selects the first column
of A. So we end up with the first column of A.
Try to picture what's going on there. Here is the R2 plane. Here is the unit circle in
purple. And we're starting off at the point 1, 0, right? We start off with 1, 0.
So right here we go one unit along the x1-axis and 0 units up the x2- axis. So start right
there at 1, 0. The first thing that happens in our linear transformation is we get multiplied
by A rot, right? That's the first thing that happens we get
multiplied by A rot then we get multiplied by A ref. If I take A rot times 1, 0, I end
up with the point square root 3 over 2 negative 1/2. Well, that's this point right here, square
root 3 over 2 minus 1/2. I've been rotated pi over 6 radians in the clockwise direction.
What happens next? The next thing that happens is we're going to get reflected about the
line x1 equals and negative x2. There's my best stab at drawing a straight line through
the origin with a negative 1 slope. And we're going to get rotated or I'm sorry,
reflected about that line. So if we do that multiplication, if we actually take this vector
times A ref we actually get the point 1/2 and negative square root 3 over 2. That reflects
us about the line x1 equals a negative x2. So this point right here-- I'm sorry, I meant
to circle it, not draw a line through it. This point right here we computed by just
taking A ref times this point right here. So starting at this point, what happens next
is the reflection operation operates on it to reflect it down there. And that operation
is described by this multiplication, which gives us this vector right here.
So you can see I've been reflected about the line x1 equals and negative x2. I've hopped
from this point and then reflected over to this point. So that's the end of our example.
We've computed the matrix representation of a linear transformation that rotates then
reflects a point. And we were able to break that down in these two pieces of reflection
operation and rotation operation. The biggest thing to be careful about when you work these
problems is that in general, A ref, A rot is not equal to A rot, A ref, right.
Matrix multiplication, the order matters almost always. So just make sure that when you write
down an operation like this that consists of a sequence of steps, you have those steps
written down in the right order.