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Math Help - linear transformation/linear combination question

  1. #1
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    linear transformation/linear combination question

    Let B = { u_1, u_2, u_3} and B' = { v_1, v_2, v_3}
    Be the two bases for R^3
    given that the transition matrix from B' to B is

    P = \begin{bmatrix}1 & -1 & 2\\0 & 1 & 2\\3 & 0 & -1\end{bmatrix}


    how can i find the u_1, u_2, u_3 as linear combinations of v_1, v_2, v_3
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  2. #2
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    Re: linear transformation/linear combination question

    I could be mistaken, but I think that v1 in it's "native basis" of B' would be written just as [1 ; 0 ; 0] (a column vector). So when you left multiply v1 by the matrix P, what you get is u1. So u1 (expressed in the B' basis, ie as a linear combo of v1 v2 v2) is just the first column of P, [1; 0 ; 3]. Similarly, u2 = [-1; 1 ; 0] and u3 = [2 2 -1].

    If you want to go the other way, just invert P. It is invertible.
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