# Homework Help: Linear transformations

1. Feb 22, 2010

### EV33

1. The problem statement, all variables and given/known data
Derive a formula for T.

T([1 1]^T)=[2 -1]^T and

T([1 -1]^T)=[0 3]^T

(...^T=transpose and T(...)=Linear Transformation
2. Relevant equations

T(c1v1+...+cnvn)=c1T(v1)+...+cnT(vn)

3. The attempt at a solution

The solutions manual's method and the method I am supposed to use for these problems.

Let u1=[1 1]^T and u2=[0 -1]^T. If x=[x1 x2]^T then x=[(x1+x2)/2]u1+[(x1-x2)/2]u2...

this is the full solution but I don't know what they are doing here. My question is where did the 1/2's come from?

It says to use the equation that have in the relavant equations part, but I don't see how that would get me where they are.

Using that I would get...

T(u1+u2)=T(u1)+T(u2)=a1u1+a2u2=a1[2 -1]^T + a2[0 3]^T

and I am stuck at this point and don't see a connection between what I have and what they have.

2. Feb 22, 2010

### Staff: Mentor

You really want to know what T does to the standard basis {[1 0]^T, [0 1]^T}. As it turns out, [1 0]^T = 1/2 * [1 1]^T + 1/2* [1 -1]^T, and
[0 1]^T = 1/2 * [1 1]^T - 1/2* [1 -1]^T.

3. Feb 23, 2010

### EV33

I don't be a pain here but I don't see the connection. The problem doesn't say anything about the standard basis.

So am I supposed to set up 2 matrices each time I have a problem like this and put e1 in the augmented side of 1, and e2 in the augmented side of the other?

4. Feb 23, 2010

### Staff: Mentor

If you know what T does to [1 0]T and [0 1]T, then you know what it does to any arbitrary vector [x y]T. The reason is that [x y]T = x* [1 0]T + y*[0 1]T.

So T([x y]T) = T(x* [1 0]T + y*[0 1]T) = x*T([1 0]T) + y*T([0 1]T).

5. Feb 23, 2010

### EV33

Thank you for all the help so far Mark 44, but I am having trouble getting this.

I know what we are trying to do here. But this process makes no sense to me.I know we are just looking for an equation that we plug x from T(X) into to get T(x). I can do these easy problems obviously without doing this process, but I am guessing these can get a lot uglier so if someone could just walk me through a full problem I think that might really help me. If someone is up for that I can post a problem or if you have a good example problem that works too.

Let's try T([1 1]^T)=[1 2 1]^T
and T([1 -1]^T)=[0 2 2]^T

Let's call the vectors being transformed u1 and u2

x=a1u1+a2u2
T(x)=a1[1 2 1]^T+a2[0 2 2]^T

then I set up a matrix
1 1 1
1 -1 0

1 1 0
1 -1 1

then I get the x1 and x2 of each of those to be 1/2.

So a1=[(x1+x2)/2),(2x1+2x2)/2),(x1+x2)/2]^T
a2=[(x1+x2)/2),(x1-x2)/2),(x1-x2)/2)]^T

Is this right so far?

If so how do I finish the problem up?

6. Feb 23, 2010

### Staff: Mentor

No, that's not it. We aren't looking for an equation we can plug in. We don't know T(x). What we would like is some matrix A such that T(x) = Ax.

All we know are T(u) and T(v) for some arbitrary vectors u and v. We would like to know what T(x) is for any old vector x. We're trying to figure out what the transformation does to a basis, because if we know that, we know what it will do to any vector. This is what I said in post #4.
OK. By inspection, I can see that (1/2)(u1 + u2) = [1 0]T, and also that (1/2)(u1 - u2) = [0 1]T. This shouldn't be too much of a surprise, since u1 and u2 are the same vectors as in the first problem.
No, you can't say that. You don't know what T(x) is - that's really what the problem wants you to find. Also, let's leave x out of things. Here's what we know.
T([1 0]T) = T((1/2)(u1 + u2)) = (1/2)T(u1) + (1/2)T(u2) = [1/2 2 3/2]T

Also, T([0 1]T) = T((1/2)(u1 - u2)) = (1/2)T(u1) - (1/2)T(u2) = [1/2 0 -1/2]T

In the work above, I used what you gave me for T(u1) and T(u2).

Now, we're ready to define T(x), where x is the vector [x1 x2]T.

T(x) = T([x1 x2]T) = T(x1*[1 0]T + x2*[0 1]T)
= x1*T([1 0]T) + x2*T([0 1]T)
= x1* (what?) + x2*(what?)

This defines T(x).

As a side note, T([1 0]T) and T([0 1]T) can be used to define the columns of a 3 x 2 matrix that I'll call A.

You can also define the transformation T(x) using the matrix:
T(x) = Ax

For this problem, A turns out to be
[1/2 1/2]
[2 0 ]
[3/2 -1/2]