4.  

1. (b) Hint: The vector x is in both R(A) and $N\left(A^T\right)$

5. Hint: If ϕ is the angle between Qx and Qy and θ is the angle between x and y, show that $\text{cos }\varphi =\text{cos }\theta$.

9.  

1. (a) Hint: Show that if ${\mathbf{q}}_j$ is the jth column vector of Q then ${\mathbf{q}}_j$ is in $N\left(A^T\right)$ and show that $P\mathbf{\text{x = 0}}$ for any vector x in $N\left(A^T\right)$.

2. (b) Hint: If $\{{\mathbf{\text{u}}}_1,{\mathbf{\text{u}}}_2,{\mathbf{\text{u}}}_3,{\mathbf{\text{u}}}_4\}$ is an orthonormal basis for R(A) and $\{{\mathbf{\text{u}}}_5,{\mathbf{\text{u}}}_6,{\mathbf{\text{u}}}_7\}$ is an orthonormal basis for $N\left(A^T\right)$ and we set

   $$U_1=({\text{u}}_1,{\text{u}}_2,{\text{u}}_3,{\text{u}}_4)U_2=({\text{u}}_5,{\text{u}}_6,{\text{u}}_7)$$

   then $U=(U_1,U_2)$ is an orthogonal matrix.

11.  

1. (b) Hint: The vectors cos x and sin x are orthogonal, so you can use the Pythagorean Law.