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## Using matrices to transform the plane

# Using matrices to transform the plane: Mapping a vector

## Video transcript

- [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.