Let u1, ...,uk be an orthonormal basis for the subspace W Rm. Let A = (u1, u2

Let u1, ...,uk be an orthonormal basis for the subspace W Š‚ Rm. Let A = (u1, u2 ... uk) be the m × k matrix whose columns are the orthonormal basis vectors, and define P = AAT to be the corresponding projection matrix.
(a) Given v ˆŠ Rn, prove that its orthogonal projection w ˆŠ W is given by matrix multiplication: w = P v.
(b) Write out the projection matrix corresponding to the subspaces spanned by
(i)

(ii)

(iii)

(iv)

(v)

(c) Prove that P = PT is symmetric.
(d) Prove that P2 = P. Give a geometrical explanation of this fact.
(e) Prove that rank P = k.

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