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Math Help - find the matrix for ProjL with respect to the basis

  1. #1
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    find the matrix for ProjL with respect to the basis

    Let L be the line in R3 spanned by v1=[1 1 1]
    (a)Find a basis {v1, v2} for the plane perpendicular to L.
    (b) Let ProjL be the projection onto the line L. Find the matrix for ProjL with respect to the basis B.

    Sol: We need to find 2 vectors that span the above plane, that is, we need to find the null space of the matrix. I got [1 1 1 0], which is already in row reduced echelon form. After doing some algebra, I got span([0 -1 1], [-1 0 1]). Therefore, the basis of R3 is {[0 -1 1], [-1 0 1]} right? But I don't get part (b). Help!
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  2. #2
    Senior Member Tinyboss's Avatar
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    Pick two linearly-independent vectors w,v which each have zero dot product with [1,1,1]. Then (w,v) is a basis for the plane perpendicular to [1,1,1], and (w,v,[1,1,1]) is a basis for R3. Then projection onto the line L sends w and v to zero, and [1,1,1] to itself, which is all you need to know to write its matrix under that basis.
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