Chapter 6.3, Problem 21E

In Exercises 21 and 22, all vectors and subspaces are in ℝⁿ. Mark each statement True or False. Justify each answer.

21.

1. a. If z is orthogonal to u₁ and to u₂ and if W = Span {u₁, u₂}, then z must be in W^⊥.
2. b. For each y and each subspace W, the vector y − proj_W y is orthogonal to W.
3. c. The orthogonal projection ŷ of y onto a subspace W can sometimes depend on the orthogonal basis for W used to compute ŷ.
4. d. If y is in a subspace W, then the orthogonal projection of y onto W is y itself.
5. e. If the columns of an n × p matrix U are orthonormal, then UU^Ty is the orthogonal projection of y onto the column space of U.