Using matrices to transform the plane
Using matrices to transform the plane: Mapping a vector
- [Instructor] Let's say that we have the vector three, two. We know that we can express this as a weighted sum of the unit vectors in two dimensions or we can view it as a linear combination. And you could view this as three times the unit vector in the X direction which is one zero plus two times the unit vector in the Y direction, which is zero, one. And we can graph three, two 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. And then we have plus two unit vectors in the Y direction. So one and then two, and then we know where our vector is or what it would look like. The vector three, two would look like this. Now let's apply a transformation to this vector. And so let's say we have the transformation matrix. I'll write it this way. Two, one, two, three. 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. And so 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 three one, zeros, we are now going to have three two, ones. Instead of having two zero, ones, we're now going to have two two, threes. 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 three times instead of one, zero, it's going to be times two, one vectors. That's the image of the one, zero unit vector under this transformation. And then we're gonna say plus two instead of zero, one, we're gonna look at the image under the transformation of the zero, one vector, which the transformation matrix gives us and that is the two, three, vector. Two, three, and we can graph this. If we have three two, ones and two two, threes, what I could do is overlay this extra grid to help us. So this is two, one, that's one to one, that is two two, ones going here. And then we have three two, ones right over here. So there's three two, ones. Let me do this in this color. This part right over here is going to be this vector. The three two, ones is going to look like that. And then to that, we add two two, threes. So this is going to... Let's see, two and then three so this is going to be one two, three, and then we have two two, threes. So we end up right over there. And so let me actually get rid of this grid so we can see things a little bit more clearly. And so we have here in purple, we have our original three, two vector and now the image is going to be three two, ones plus two two, threes so the image of our three, two vector under this transformation is going to be this vector that I'm drawing right here. And it looks when I eyeball it, it looks like it is the 10, nine vector and we can verify that by doing the math right over here. So let's do that. This is going to be equal to three times two is six, three times one is three. And we're going to add that to two times two is four, two times three is six. And indeed you add the corresponding entry, six plus four is 10 and three plus six is nine. 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 factors. And we saw that visually and we verified that mathematically.