Question

All vectors and subspaces are in R”. Mark each statement True or False. Justify each answer. Complete parts (a) through (e) below. a. If W is a subspace of R" and if y is in both W and wt, then y must be the zero vector. If v is in W, then projwv = Since the wt component of v is equal to v the w+ component of v must be A similar argument can be formed for the W component of v based on the orthogonal projection of v onto the subspace wt. Thus, v be 0. y.u1 b. In the Orthogonal Decomposition Theorem, each term in ý= + yup up is itself an orthogonal projection of y onto a subspace of W. up up uyu, 4+ Since you u :u u; is the projection of v onto and the span of each is a V-dimensional subspace of W, each resulting projection be onto the subspace spanned by v c. If y=24 +22, where 24 is in a subspace W of R" and zz is in wt, then 24 must be the orthogonal projection of y onto W. Since the orthogonal decomposition of y into components that exist in W and wt is vz correspond to the orthogonal projection of y onto W. d. The best approximation to y by elements of a subspace of W is given by the vector y - projwy. V The says that the best approximation to y is e. If an nxp matrix U has orthonormal columns, then Uu'x= x for all x in R". Let W be the subspace spanned by the columns of U. Then UTUx = projwx. If p= n, then W be all of R", so the statement is V for all x in R". Ifp<n, then W V be all of R", so the statement is V for all x in R".

Answer 1

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