Using matrices to transform the plane: Mapping a vector | Matrices | Precalculus | Khan Academy

Let's say that we have the vector (3, 2). We know that we can express this as a weighted sum of the unit vectors in two dimensions, or we could view it as a linear combination. You could view this as (3) times the unit vector in the (x) direction, which is ((1, 0)), plus (2) times the unit vector in the (y) direction, which is ((0, 1)).

We can graph ((3, 2)) by saying, okay, we have three unit vectors in the (x) direction. This would be one right over there, that would be two, and then that would be three. Then we have plus two unit vectors in the (y) direction, so one and then two. We know where our vector is or what it would look like. The vector ((3, 2)) would look like this.

Now, let's apply a transformation to this vector. Let's say we have the transformation matrix. I'll write it this way: (\begin{pmatrix} 2 & 1 \ 2 & 3 \end{pmatrix}).

Now, we've thought about this before. One way of thinking about a transformation matrix is it gives you the image of the unit vectors. Instead of being this linear combination of the unit vectors, it's going to be this linear combination of the images of the unit vectors when we take the transformation. What do I mean? Well, instead of having (3(1, 0)), we are now going to have (3(2, 1)). Instead of having (2(0, 1)), we're now going to have (2(2, 3)).

So I could write it this way. Let me write it this way: the image of our original vector, I'll put a prime here to say we're talking about its image, is going to be (3) times instead of ((1, 0)), it's going to be times ((2, 1)). That's the image of the ((1, 0)) unit vector under this transformation. Then, we're gonna say plus (2) instead of ((0, 1)). We're gonna look at the image under the transformation of the ((0, 1)) vector, which the transformation matrix gives us, and that is the ((2, 3)) vector.

We can graph this. If we have ((3, 2)) and ((2, 2)), what I could do is overlay this extra grid to help us. So this is ((2, 1)), that's ((1, 2)). ((1, 2)) is ((2, 2)).

So, we have ((3, 2)) right over here. Let me do this in this color. This part right over here is going to be this vector. The ((3, 2)) is going to look like that. Then to that, we add ((2, 3)). So this is going to be (1, 2), and then (3). So this is going to be (1, 2, 3) and then we have ((2, 2)). So we end up right over there.

Let me actually get rid of this grid so we can see things a little bit more clearly. Here we have in purple our original ((3, 2)) vector. Now the image is going to be ((3, 2)) plus ((2, 3)).

So the image of our ((3, 2)) vector under this transformation is going to be the vector that I'm drawing right here. When I eyeball it, it looks like it is the ((10, 9)) vector. We can verify that by doing the math right over here.

So let's do that. This is going to be equal to (3 \times 2 = 6), (3 \times 1 = 3), and we're going to add that to (2 \times 2 = 4), (2 \times 3 = 6). Indeed, you add the corresponding entries: (6 + 4 = 10), and (3 + 6 = 9), and we're done.

The important takeaway here is that any vector can be represented as a linear combination of the unit vectors. Now, when we take the transformation, it's now going to be a linear combination not of the unit vectors, but of the images of the unit vectors. We saw that visually, and we verified that mathematically.