Matrices as transformations of the plane | Matrices | Precalculus | Khan Academy

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.

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: (1, 0). 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 (1, 0) vector.

Then this right over here, what would we call that? Well, that we would call the (0, 1) 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 right over here, let's call it point A. 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 (3, 1). The x coordinate is at 3, the y coordinate is 1. But if we wanted to express it in terms of a vector, we could write it as (3, 1). In the x direction, we're moving 3 from the origin, positive 3, and in the y direction, we're moving 1 to get there. 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 3 times our (1, 0) vector plus 1 times our (0, 1) vector. And you could 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. 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 (1, 0) and then the second column is (0, 1). 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 (1, 0) vector, this first blue vector. Well, we're just keeping it (1, 0), so we're not changing it. It's the same (1, 0) vector here, (1, 0) vector here. Likewise, (0, 1) vector here, (0, 1) 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 (1, 0) here, I'm going to write (2, 1) here. Instead of (0, 1) here, let me write (1, 2).

So in this transformation, what we're doing is we're turning this (1, 0) vector into a (2, 1) vector. You're going to see what I'm talking about in a second. So what does a (2, 1) vector look like? Let me do it in that color. Well, we go 2 in the x direction, 1 in the y direction, so it's going to look like this; it's going to look like that.

And then what does a (1, 2) vector look like? Well, it goes 1 in the x direction and then 2 in the y direction, so it looks like this. The way that this represents a transformation is that anything that was a weighted sum of the (1, 0) and the (0, 1) vectors originally, you can now view as a weighted sum of the (2, 1) and the (1, 2) vectors.

And so we can now think of another point A prime that's not going to be three of the (1, 0)s and one of the (0, 1)s. We can think of it as, let me write it over here, as three of the (2, 1)s plus one of the (1, 2)s. So where would that put that now? Well, we're going to do three of the (2, 1)s. 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 (1, 2) 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 point 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 0 of the orange vector, 0 of the blue vectors. So even after the transformation, it's going to be 0 of the (2, 1) vector and 0 of the (1, 2) 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.

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 (2, 1) vectors and none of the (1, 2) vectors, so two of the (2, 1)s. So if you go one, two and then none of the (1, 2)s, 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 ABC. What is it now, gotten mapped to? Well, it's now mapped to this big triangle. 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 want to connect that side from C prime to A prime.

Now you might be saying, “Sal, 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 (2x2) 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.