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Math Help - Basis

  1. #1
    Super Member Deadstar's Avatar
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    Basis

    Let B_1 = (v_1, v_2, v_3) be an ordered basis of a vector space V and let B_2 = (w_1, w_2, w_3), where;

    w_1 = v_2 + v_3, w_2 = v_1 + v_3, w_3 = v_1 + v_2

    Verify that B2 is also a basis of V and find the change of basis matrices from B1 to B2 and from B2 to B1: Express the vector av_1 + bv_2 + cv_3
    as a linear combination of w_1, w_2 and w_3

    I dont really understand basis at all but my attempted answers so far are...

    B2 is a basis of V because w_1, w_2 and w_3 are all linearly independent of V and have same dimension... maybe...

    av_1 + bv_2 + cv_3 can surely just be expressed as any multiple of {x}w_1 + {y}w_2 + {z}w_3. For example if x=y=z=1 then a=b=c=2. Am i getting this bit totally wrong? It seems to simple an answer...

    Any help with any of it would be could especially the change of basis bit.
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  2. #2
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    The matrix \frac{1}{2}\left[ {\begin{array}{rrr}{ - 1} & 1 & 1  \\ 1 & { - 1} & 1  \\    1 & 1 & { - 1}  \\  \end{array} } \right] will transform a vector in the ‘v-basis’ into a vector into the ‘w-basis’.

    That matrix is the inverse of the matrix \left[ {\begin{array}{ccc}   0 & 1 & 1  \\   1 & 0 & 1  \\   1 & 1 & 0  \\ \end{array} } \right].
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