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Linear Transformations, Linear Algebra Question

  1. May 10, 2015 #1
    Hi can any one give me some hints with this question thanks

    A = \begin{pmatrix} 3 & -2 &1 & 0 \\ 1 & 6 & 2 & 1 \\ -3 & 0 & 7 & 1 \end{pmatrix}

    be a matrix for T:ℝ4→ℝ3 relative to the basis

    B = {v1, v2, v3, v4} and B'= {w1, w2, w3}

    v1 = \begin{pmatrix} 0 \\ 1 \\ 1 \\ 1 \end{pmatrix}
    v2 = \begin{pmatrix} 2 \\ 1 \\ -1 \\ -1 \end{pmatrix}
    v3 = \begin{pmatrix} 1 \\ 4 \\ -1 \\ 2 \end{pmatrix}
    v4 = \begin{pmatrix} 6 \\ 9 \\ 4 \\ 2 \end{pmatrix}
    w1 = \begin{pmatrix} 0 \\ 8 \\ 8 \end{pmatrix}
    w2 = \begin{pmatrix} -7 \\ 8 \\ -1 \end{pmatrix}
    w3 = \begin{pmatrix} -6 \\ 9 \\ 1 \end{pmatrix}

    a- Find [T(v_1)]B' , [T(v_2)]B' , [T(v_3)]B' and [T(v_4)]B'.
    b- Find T(v1), T(v2), T(v3) and T(v4).
    c- Find a formula for T( \begin{pmatrix} x_1 \\ x_2 \\ x_3 \\ x_4 \end{pmatrix} )

    Attempt solution for part (a)
    [T(v)]B' = [T]B→B' × [v]B
    If this is right I dont know how to use it.
    Also i tried drawing a diagram but i think i have to use the diagram to find a formula for T in part (c).

    Thanks
     
    Last edited by a moderator: May 11, 2015
  2. jcsd
  3. May 11, 2015 #2

    Fredrik

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    If I understand your notation correctly, it's correct. I would write ##[Tv]_{B'}=[T]_{B',B}[v]_B##. The right-hand side is the product of two matrices, so the next step should be to multiply the matrices.
     
  4. May 11, 2015 #3
    but how do i get [T]B' ,B ? like [T(v1)]B' ,B I know how to do it with polynomials but I have no idea with matrices.
     
  5. May 11, 2015 #4

    Fredrik

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    ##[T]_{B',B}## is the matrix you included at the start of your post.
     
  6. May 14, 2015 #5
    Please can you explain why it is that ? Thanks
     
  7. May 14, 2015 #6

    Fredrik

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    There isn't much to explain. By definition ##[T]_{B',B}## denotes the matrix of T with respect to the pair of bases B and B' (B for the domain and B' for the codomain), and that's what you said that A is. So we have ##A=[T]_{B',B}##.

    The https://www.physicsforums.com/threads/matrix-representations-of-linear-transformations.694922/ [Broken] may be useful.
     
    Last edited by a moderator: May 7, 2017
  8. May 14, 2015 #7
    Oh kkkkkkkkkk got it. Thanks so much.
     
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