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Homework Help: Linear transformations

  1. Feb 22, 2010 #1
    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. jcsd
  3. Feb 22, 2010 #2

    Mark44

    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.
     
  4. Feb 23, 2010 #3
    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?
     
  5. Feb 23, 2010 #4

    Mark44

    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).
     
  6. Feb 23, 2010 #5
    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?
     
  7. Feb 23, 2010 #6

    Mark44

    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]

     
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