Using matrices to transform the plane: Composing matrices | Matrices | Precalculus | Khan Academy

So what I have here is two different transformation matrices. What we're going to think about in this video is: can we construct a new matrix that's based on the composition of these transformations?

Or, a simpler way of saying that is a new transformation that's based on applying one of these transformations first and then the other one right after that.

So first, let's just review what's going on. If we have some random vector here, (a, b), we know that we could view this as (a) times the ((1, 0)) vector, the unit vector in the x direction, plus (b) times the ((0, 1)) vector, which is the unit vector that goes in the vertical direction.

Now, if you were to apply this transformation, capital (A), here, it tells you instead of using ((1, 0)) and ((0, 1)), use these two columns instead. So if you were to apply the transformation here, I guess we could call it (A) for (b'), that is going to be, if you apply the capital (A) transformation matrix, it's going to be (a) times not ((1, 0))—you use ((0, 5)) instead. And then plus (b) times not ((0, 1))—you use ((2, -1)) instead.

So that's just a little bit of review, but what we're going to think about in this video is: what would be the transformation matrix for the composition? And I could write that as (B(A)) right over here, and you might recognize this from function notation where essentially it's saying you would apply the function (A) first and then whatever the output of that is, you would then input that into (B) and you would get the output of that.

And that makes sense because you can view transformation matrices really as functions—functions that map points on the coordinate plane. So in this situation, what would be the transformation matrix that is a composition of these two?

Pause this video and think about that.

All right, well, what would happen is we would first transform any point using these two vectors: the ((0, 5)) and the ((2, -1)) because that's the first transformation we do. And then we would apply this transformation to whatever the resulting vector is. Now, that seems pretty involved, and we don't want to write it in terms of (a) or (b)’s; we just want to write it in terms of a transformation matrix.

So one way to think about it is we can transform each of these vectors that you have in matrix (A), because remember that says what do you turn the vectors ((1, 0)) and ((0, 1)) into. So if we transform ((0, 5)) using the matrix (B) and if we transform ((2, -1)) using the matrix (B) and we put them in their respective columns, we should have the composition of this.

So let me write it this way and create a little bit of space.

So let's say that the composition (B(A)) is equal to, all right, a big two by two matrix right over here. The first thing we can do is apply transformation matrix (B) to the purple column right over here. And what is that going to tell us? Well, that's going to be (0) times ((-3, 1)). So let me write it that way; it's going to be (0) times ((-3, 1)) plus (5) times ((0, 4)).

And this is going to give us a two by one vector right over here, so you can view it as filling up the first column of this transformation, this composition, I guess you could say. And then let's think about this second vector right over here, ((2, -1)). If you transform that using (B), what are you going to get? You're going to get (2) times ((-3, 1)).

So, I'll write it here: (2) times ((-3, 1)) plus (-1) times ((0, 4)). And this doesn't look like a matrix just yet, but if you work through it, it will become a matrix. For example, if I multiply, well, (0) times all of this is going to be (0) and then (5) times (0) is going to be, let me just write it this way: this would turn into (5) times (0) is (0) and (5) times (4) is (20).

And then this matrix right over here, (2) times ((-3, 1)) is going to be ((-6, 2)), and then we have minus ((0, 4)). And now, if we wanted to write this clearly as a two by two matrix, this would be equal to—and we get a little bit of a drum roll here—the first column is ((0, 20)) and then the second column is going to be, let's see, ((-6 - 0)) is still (-6), and ((2 - 4)) is (-2).

And we're done! We have just created a new transformation matrix that's based on the composition (B(A)). So if you apply transformation (A) first to any vector and then apply transformation (B) to whatever you get there, that is equivalent to just applying this one two by two transformation matrix (B(A)).