Question

(1 point) Are the following statements true or false? ? 1. The best approximation to y by elements of a subspace W is given by the vector y - projw(y). ? 2. If W is a subspace of R" and if V is in both W and Wt, then v must be the zero vector. ? 3. If y = Z1 + Z2 , where z is in a subspace W and Z2 is in W+, then Z, must be the orthogonal projection of y onto W. ? 4. Let uj, u,, ... , u, be an orthonormal basis of a vector space V. In the Orthogonal Decomposition Theorem, each term p = (b, u Su + ... + (b, u, )u, is itself an orthogonal projection of b onto a subspace of V. ? 5. If an n x p matrix U has orthonormal columns, then UUTx = x for all x in R".

Answer 1

1. False. 2. True. 3. True. 4. True. 5. True. Explanations: 1. The best approximation is projw (7) 2. The only vector orthogonal to itself is zero vector. 3. By definition of projection, we get projw(T)) = projw (71 +22)) = projw (71) + projw(22) = 21 + 7 = 21 4. It is an orthogonal projection to the subspace V itself. 5. VUT = 1