Let's evaluate each statement:

A. True. If an $n \times p$ matrix $U$ has orthonormal columns, then $UU^T$ is the orthogonal projection matrix onto the column space of $U$. Therefore, $UU^T x = x$ for all $x$ in $\mathbb{R}^n$ that are in the column space of $U$.

B. False. In the Orthogonal Decomposition Theorem, the term $\hat{y}$ is the orthogonal projection of $y$ onto the subspace spanned by the vectors $u_1, \ldots, u_p$, not onto a subspace of $W$.

C. False. If $y = z_1 + z_2$, where $z_1$ is in the subspace $W$ and $z_2$ is in $W^\perp$, then $z_1$ is the orthogonal projection of $y$ onto $W$.

D. False. The best approximation to $y$ by elements of a subspace $W$ is the projection of $y$ onto $W$, denoted by $\text{proj}_W(y)$, not $y - \text{proj}_W(y)$.

E. True. If $W$ is a subspace of $\mathbb{R}^n$ and $v$ is in both $W$ and $W^\perp$, then $v$ must be the zero vector because the only vector in both a subspace and its orthogonal complement is the zero vector.

Thus, the true statements are A and E.