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Projection onto subspace along subspace

  1. Apr 14, 2009 #1
    1. The problem statement, all variables and given/known data
    Find a projection [matrix] E which projects R2 onto the subspace spanned by (1,-1) along the subspace spanned by (1,2).


    2. Relevant equations
    [tex]P = \frac{a a^{T}}{a^{T} a}[/tex]


    3. The attempt at a solution
    Computing P...
    [tex]P = \[ \left( \begin{array}{ccc}
    \frac{1}{2} & -\frac{1}{2}\\
    -\frac{1}{2} & \frac{1}{2} \end{array} \right)\][/tex]

    Let D be a change of basis matrix from the standard basis to the basis B = {(1,-1), (1,2)}
    [tex]D = \[ \left( \begin{array}{ccc}
    1 & 1 \\
    -1 & 2 \end{array} \right)\][/tex]

    E = D-1PD?

    E2 = D-1PDD-1PD = D-1P2D = D-1PD, so it passes that test for being a projection.

    Is E the projection talked about in the question?
     
    Last edited: Apr 15, 2009
  2. jcsd
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