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Math Help - Matrices for linear transformation

  1. #1
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    Matrices for linear transformation

    Here is the problem:

    Given the linear transformation T: R3 -> R3, such that T(V)=kV, where k is a constant. Find the matrix(A) of T reltve to th bases B and B' for the following:

    a) B=standard basis of R3=B'
    b) B=standard basis of R3, B'={(1,0,0),(0,1,1),(0,1,1)}
    c) Verify [T(1,2,3)]B' = A[(1,2,3)]B

    Ok, here is what I did so far:

    For part a): A=[ 1 0 0
    0 1 0
    0 0 1] this is the standard basis for basis of R3

    and therefore I get T(v)=[k 0 0
    0 k 0
    0 0 k]. It seems quite weird to me. But I could not tell where is wrong.

    For part b): since the standard basis for B is{(1,0,0) (0,1,0) (0,0,1)}, and B'=={(1,0,0),(0,1,1),(0,1,1)}

    T(1,0,0) = (k,0,0)
    T(0,1,0)= (0,k,0)
    T(0,0,1)=(0,0,k) for each of these three, write them in terms of B', however, unlike the usual problem dealing with numbers, so I am stuck on this. What to do next?
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  2. #2
    Super Member Gamma's Avatar
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    Iowa City, IA
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    B' is not a basis for \mathbb{R}^3, those last 2 elements are the same thus they are clearly not linearly independent.

    But for your part a) one you are correct.

    I suspect something has been copied down incorrectly one place or another.

    In any case, the idea is correct for your part b so far. What you need to do is see what the image of the first basis element is as you have done, it is (k,0,0). Then see how to represent it in terms of the other basis. That is precisely the problem, that B' set does not span \mathbb{R}^3 so this will not be possible in this case and this is why you are having trouble here.
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  3. #3
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    Thanks for your reply. I am gonna double check with my instructor, but I believe that is what he gave to us.
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