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    Matrix representation

    Compute the standard matrix representation of the orthogonal projection of R^4 onto the subspace spanned by {(1,1,1,1),(1,2,3,4)}.
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    Quote Originally Posted by matty888 View Post
    Compute the standard matrix representation of the orthogonal projection of R^4 onto the subspace spanned by {(1,1,1,1),(1,2,3,4)}.
    Step 1. Find an orthonormal basis for the subspace, e.g. \textstyle\{\frac12(1,1,1,1),\,\frac1{\sqrt{20}}(-3,-1,1,3)\}.

    Step 2. In R^n, the projection onto the one-dimensional subspace spanned by a single unit vector (a_1,a_2,\ldots,a_n) has a_ia_j as its (i,j)-entry. So for example the projection onto the subspace spanned by \textstyle\frac1{\sqrt{20}}(-3,-1,1,3) is \frac1{20}\begin{bmatrix}9&3&-3&-9\\ 3&1&-1&-3\\ -3&-1&1&3\\ -9&-3&3&9\end{bmatrix}. Now do the same thing for the projection onto the subspace spanned by (1,1,1,1).

    Step 3. If you have an orthonormal set of vectors then the projection onto the subspace spanned by them is the sum of the projections onto the one-dimensional subspaces spanned by the basis vectors. So the answer to the question is the sum of the two matrices from the previous paragraph.
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