Exercise 7

Determine whether the following statements are true or false and explain your thinking.

1. If the columns of $Q$ form an orthonormal basis for a subspace $W$ and $\wvec$ is a vector in $W$, then $QQ^T\wvec = \wvec$.

2. An orthogonal set of vectors in $\real^8$ can have no more than 8 vectors.

3. If $Q$ is a $7\times5$ matrix whose columns are orthonormal, then $QQ^T = I_7$.

4. If $Q$ is a $7\times5$ matrix whose columns are orthonormal, then $Q^TQ = I_5$.

5. If the orthogonal projection of $\bvec$ onto a subspace $W$ satisfies $\bhat = \zerovec$, then $\bvec$ is in $W^\perp$.