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Current time:0:00Total duration:8:12

- [Instructor] In this
video, we're going to explore how a two by two matrix can be interpreted as representing a transformation
on the coordinate plane. So let's just start with some examples or some conceptual ideas. So the first conceptual idea is that any point on our coordinate plane here and this of course is our X
axis and this is our Y axis, can be represented by a
combination of two vectors. You could have this
vector right over here. That goes exactly one unit
in the horizontal direction. We can represent that as a
vector like this, one, zero, when you write a vector
vertically like this the convention is that the top number here is what we're doing in the X direction. And then the bottom number, the zero is what we're doing
in the vertical direction or the Y direction, so this
is the one, zero vector. And then this right over
here, what would we call that? Well that we would call
the zero, one vector because it doesn't go at
all in the X direction. And it only goes up
one in the Y direction. Now just to feel good that any
point on the coordinate plane can be represented as a
weighted sum of these. Let's just pick a point at random. Let's say this point, all right over here, let's call it point A. And you could represent that as a vector that looks something like this, I'll do it as a dotted line, but this could be represented as a positional vector like that. And of course, if we're
thinking about it in coordinates we would just say this is at
the coordinate three comma one. The X coordinate is a three,
the Y coordinate is one. But if we wanted to express
it in terms of a vector, we could write it out as three, one, the X direction we're moving
three from the origin, positive three and the Y direction we're moving one to get there. And you could see that
we can represent this as a weighted sum of these two vectors. We can write this as, this is
the same thing as three times our one, zero vector, one, zero plus one times our zero
one, vector zero one vector. And you can see it visually, this yellow vector that points
to point a right over there. You can have three of this
vector, one, two, and then three. And then one of the orange vector. Now I said that I would
explain how two by two matrices can represent a transformation. And the way that you
could think about it is if I have a two by two
matrix that looks like this, so let me just draw the matrix, where the first column is one, zero and then the second column is zero, one. This just tells you what to
do with these two vectors. I know this might be a little
bit confusing at first, but let's just walk through it together. So the way I've represented it, this first column says
what is the transformation you want to apply to
this one, zero vector, this first blue vector? Well, we're just keeping it one zero. So we're not changing it, it's the same. One, zero vector here
one, zero vector here. Likewise, zero, one vector
here, zero, one vector here. So this two by two matrix
actually represents what's sometimes known as
the identity transformation. It maps any point on the
coordinate plane back to itself. It doesn't change the points, but I'm showing this because
now I'm going to show a two by two matrix that represents a non identity transformation. For example, let me draw the matrix again. So let's say I have the matrix, instead of one, zero here I'm
going to write a two, one here instead of a zero, one here
let me write a one, two. So in this transformation,
what we're doing is we're turning this one, zero
vector into a two, one vector. You're going to see what I'm
talking about in a second. So what does a two, one vector look like? Let me do it in that color. Well, we go two in the X
direction one in the Y direction. So it's going to look like this, it's going to look like that. And then what does a one,
two vector look like? Well, it goes one in the X direction and then two in the Y direction,
so it looks like this. And the way that this
represents a transformation is that anything that was a weighted sum of the one, zero and the
zero, one vectors originally, you can now view as a
weighted sum of the two, one and the one, two vectors. And so we can now think
of another point A prime, that's not going to be
three of the ones zeros and one of the zero ones. We can think of it as,
let me write it over here, as three of the two, ones plus one of the one twos,
one of these one, two. So where would that put that now? What we're going to go,
go three of the two, ones. So this is one of them,
this is two of them and then this is three of them. And then I'm going to have
one of the one, two vectors, this orange vector right over here. And so I'm going to have one of those. And so that will take me to A prime. This is my new point
after the transformation, I go to A prime. So I've gone from this
point A to this A prime. And this two by two matrix is
telling us how to transform. Now we can do that with multiple points. Now, if I talk about a
point here at the origin that was originally at the origin point B, well that's zero of the orange vector, zero of the blue vectors. So even after the transformation, it's going to be zero
of the two, one vector and zero of the one, two vectors. So it's just going to stay in place. It's just going to map to itself. So B is equal to B prime, and we could also imagine another point. Let's say right over here,
let's call that point C. Well point C is originally
two of the blue vectors and none of the orange vectors. So after the mapping, it'll
be two of the two, one vectors and none of the one, two vectors. So two of the two ones. So if you go one, two, and
then none of the one, twos you're going to get C
prime right over there. And so notice, if originally
you had a triangle between A, B and C, let
me draw it like this. So originally you had
this triangle A, B, C what is it now gotten mapped,
what is it now mapped to? Well, it's now mapped to big triangle. I'll do my best to draw
it relatively straight so we land on target. So that's that side. And then we have this side
going from B prime to C prime and then we wanna connect that
side from C prime to A prime. Now you might be saying, so how do you know that the
lines map onto other lines? How do you know this transformation didn't all of a sudden make
this line squiggly or zigzag? And that's one of the
interesting properties of the type of transformation
we're talking about. A two by two matrix will
represent a linear transformation. And there's two ways to think
about it in this context, a linear transformation
will always map the origin onto itself and it will always
map a line onto another line, it won't turn that line into a curve, or it won't make it zigzag somehow. Now the last thing you
might be wondering is, Hey what about all these transformations we had from geometry, these
similarity transformations, things like rotations
reflections, dilations, can you do those with matrices? And the simple answer is yes, you can do them as long as
you keep the origin in place. And you can actually
using a two by two matrix come up with a whole series of
other linear transformations that are much more, let's call it exotic, than just the rotations,
reflections and dilations.