Working with matrices as transformations of the plane | Matrices | Precalculus | Khan Academy

In a previous video, I talked about how a two by two matrix can be used to define a transformation for the entire coordinate plane. What we're going to do in this video is experiment with that a little bit and see if we can think about how to engineer two by two matrices to do some of the transformations that you might be familiar with, like rotations, dilations, or reflections.

So, this is a website run by the University of Texas: web.ma.utexas.edu, and you have the URL here. I encourage you to go there and play around with it yourself. What I have here is I have our two vectors, which any point on our coordinate axis can be defined by some combination of these two vectors. This in red here is the vector 1 0. It goes one in the x direction, 0 in the y direction, and you can see that is this first column right over here in this identity matrix.

This blue vector right over here, this is the vector 0 1, which is the second column in this identity matrix. It goes 0 in the x direction and then 1 in the y direction. Now, the way to engineer a transformation is to say, well, what would that transformation do to these two vectors, and then change the numbers accordingly.

So for example, let's say that we wanted to have a reflection about the x-axis. If you did a reflection about the x-axis, this red vector would not change; it would stay 1 0. But what would happen to this blue vector? Instead of being 0 1, it would be 0 negative 1. So, in this transformation matrix, if I go from the identity matrix here, but instead of 0 1, I now put a negative 1 here, and when I press enter, this should flip this blue vector over the x-axis and essentially flip everything else with it.

So let's try that out. I'm going to press enter, and there you have it! That cute little golden retriever is now flipped over! So, that met our intuition. Now, let's go back to what we were doing before. So, that's a reflection, and you could think about what would you do if you wanted to flip the other way, across the y-axis.

Now, what about a dilation? What if we wanted to shrink everything by a factor of two? How do you think we would modify this matrix to do that? Pause this video and think about that. Well, if we want to scale everything down, what we would want is each of these vectors, especially just by a factor of 2; we'd want each of these vectors to be half as long.

So instead of 1 0 and 0 1, we would do 0.5 0 and 0.5. Let me press enter and see what happens. There you go! It indeed worked, and really this should have shown this red vector get smaller and this blue vector get smaller, but hopefully you get the idea.

So let me go, or maybe they just want to always show what we could kind of call unit vectors. But let's go back to the original, and now let's think about a rotation. This is an interesting one. Pause this video and think about how you would rotate it if you wanted to rotate this clockwise by 90 degrees.

All right, if you rotate clockwise by 90 degrees, this red vector is no longer 1 0; it would become 0 negative 1. So let me write that down: 0 negative 1, and the blue vector would then go to where this red vector is, and it would become 1 0. So let's see if we did it the right way. I'm going to click enter, and there you go! We got our 90 degree rotation.

I just gave you some examples of how you can do a pure rotation, a pure dilation, or pure reflection, but you can imagine you can also do combinations of them by manipulating this matrix accordingly. I encourage you to play around; you can do some exotic transformations if you want.

Let's see what happens if I make this a one. Press enter. Oh, that's interesting! What happens if I then make this a two? Oh, that's interesting! So, notice you can do all sorts of really interesting linear transformations.

Just as a reminder, a linear transformation is one where the origin always maps to itself and any lines are mapped to other lines. Not necessarily the same line, but whatever it gets mapped to will still be aligned.