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Math Help - Projection, Linear transformation, maybe change of basis.

  1. #1
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    Projection, Linear transformation, maybe change of basis.

    This is my problem.

    Let V=Span([1,0,2,1],[0,1,-1,1])  \subset R^4 . Find the standard matrix for the linear transformation  proj_{V}:R^4 \rightarrow R^4 .

    Here is my thinking.
    Setting the matrix  P=A(A^\top A)^{-1} A^\top where A= \left[\begin{array}{cc}1&0\\0&1\\2&-1\\1&1\end{array}\right] . This would be the transformation matrix with respect to the given basis of V. Normally, the standard matrix T would be  T=APA^{-1} but A cannot be  \left[\begin{array}{cc}1&0\\0&1\\2&-1\\1&1\end{array}\right] because that is not a square matrix and obviously not invertible. What do I do?

    Maximum explanation please because I am still trying to wrap my head around these concepts. Thanks!!!
    Last edited by BrianMath; December 1st 2010 at 05:17 PM.
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  2. #2
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    There haven't been any views yet so I doubt anyone cares but I figured it out. The question was in the section of my book called Change of Basis so I did the problem using that, only to find out that a lot cancels and the answer comes out to be the matrix P that I denote above. You have to find two vectors that span V perp so that is a new basis of R^4, but when you apply the linear transformation and the change of basis formula, they cancel out.
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