Answer
a. True
b. True
c. False
d. True
e. True
Work Step by Step
a. z is orthogonal to 2 vectors that span the subspace W, so is also orthogonal to any linear combination of these two vectors
b. The projection of y onto W is in W so y minus a vector in W is always perpendicular to W. True by the Orthogonal Decomposition Theorem and the fact that $y-proj_Wy$ is the orthogonal complement.
c. The orthogonal projection of y onto a subspace W only depends on W and not on any particular basis.
d. True
e. $UU^Ty = proj_Wy = (y\cdot u_1)u_1 + ... + (y\cdot u_n)u_n$, where
$u_1$ to $u_n$ are orthonormal.
