Exercise 9

Explain why the following statements are true.

1. If $Q$ is an orthogonal matrix, then $\det Q = \pm 1$.

2. If $Q$ is a $8\times 4$ matrix whose columns are orthonormal, then $QQ^T$ is an $8\times8$ matrix whose rank is 4.

3. If $\bhat$ is the orthogonal projection of $\bvec$ onto a subspace $W$, then $\bvec-\bhat$ is the orthogonal projection of $\bvec$ onto $W^\perp$.