Please include breif explanation Let a Rn \\ {0}, and let P = be the projection

Question

Please include breif explanation

Let a Rn {0}, and let P = be the projection matrix onto the line Span ({a}). Show that P2 = P. Assume n > 1. Is the matrix P in part (a) invertible? Please explain. Find the projection matrix P1 onto the line through a = (1, 3) and also the matrix P2 that projects onto the line perpendicular to a. (b) Compute P1 + P2 and P1 P2. Explain why you expect to get these answers. Find the projection matrix P onto the subspace spanned by a1 = (1, 0, 1) and a2 = (1, 1, -1).

Explanation / Answer

1)

(a)
P^2=(aa^T)(aa^T)/(a^Ta)^2=a(a^Ta)a^T/(a^Ta)^2=(a^Ta)aa^T/(a^Ta)^2=aa^T/(a^Ta)=P

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