(orthogonal projection using general basis) let w be a subspace of r n with basis ( 1 ; ; k ), and let a be the matrix whose column vectors are these 0 i s, i. e. a = [ 1 k ]. then for any vector v 2 r n, show the orthogonal projection from v to w is projw (v) = a a t a 1 a t v (1) (hint: let the projection of v to w be a vector c1 1 + + ck k = [ 1 k ] 2 6 4 c1 . . cn 3 7 5 = a 2 w; where = 2 6 4 c1 . . cn 3 7 5 2 r k . using the orthogonal relation at (v a ) = 0 to önd the coe¢ cient vector ). remark: when w is a 1-dimensional subspace spanned by a vector , (1) becomes the familiar projection formula projw (v) = h ; vi h ;