9. (15 pt) Multiple choice questions and fill-in-the-blank questions. NO need to justify your answer. No credit for unanswered questions. Incorrect answers will deduct the credit until zero point is earned in this section.

(a) (correct: 3; incorrect: -1; unanswered: 0) Let $W$ be a subspace of $\mathbb{R}^n$ and $W^\perp$ denotes its orthogonal complement. If $W_1$ is a subspace of $\mathbb{R}^n$ such that $x \in W_1$, then $x^T u = 0$ for all $u \in W^\perp$. Justify whether the following statements are true or false.

(i) $\dim(W_1^\perp) \le \dim(W^\perp)$

(ii) $\dim(W_1^\perp) \le \dim(W)$

(iii) $\dim(W_1^\perp) \ge \dim(W)$

(iv) $\dim(W_1^\perp) \ge \dim(W^\perp)$

(b) (correct: 3; incorrect: -1; unanswered: 0) Let $A$ be a $7 \times 5$ matrix with $\text{rank}(A) = 5$. Justify whether the following statements are true or false.

(i) There exists at least one $\mathbf{b} \in \mathbb{R}^7$ such that $Ax = \mathbf{b}$ has infinite number of least square solutions.

(ii) For any $\mathbf{b} \in \mathbb{R}^7$, $Ax = \mathbf{b}$ has infinite number of solution.

(iii) There exists at least one $\mathbf{b} \in \mathbb{R}^7$ such that $Ax = \mathbf{b}$ has a unique least square solution.

(iv) For any $\mathbf{b} \in \mathbb{R}^7$, $Ax = \mathbf{b}$ has a unique solution.

(c) (correct: 3; incorrect: -1; unanswered: 0) Let $\hat{B} = \begin{bmatrix} (i) \\ (ii) \end{bmatrix} \in \mathbb{R}^2$ be the least-squares solution that minimizes $\|Y - XB\|^2$ where

$X = \begin{bmatrix} 2 & 0 \\ -1 & 1 \\ 0 & 2 \end{bmatrix} \text{ and } Y = \begin{bmatrix} -1 \\ 0 \\ 1 \end{bmatrix},$

Find the value for (i) and (ii).

(d) (correct: 3; incorrect: -1; unanswered: 0) Let $\begin{bmatrix} (i) \\ (ii) \end{bmatrix}$ be the orthogonal projection of $\mathbf{b}$ onto $\mathbf{u}$, where

$\mathbf{b} = \begin{bmatrix} -24 & 2 \\ -10 & -10 \end{bmatrix} \text{ and } \mathbf{u} = \begin{bmatrix} 2 \\ -10 \end{bmatrix},$

Find the value for (i) and (ii).

(e) (correct: 3; incorrect: -1; unanswered: 0) Let $\begin{bmatrix} (i) \\ (ii) \\ (iii) \end{bmatrix}$ be the closest point to $\mathbf{y}$ in the subspace $W$ spanned by $\mathbf{u}_1$ and $\mathbf{u}_2$, where

$\mathbf{y} = \begin{bmatrix} 12 \\ -1 \\ 2 \end{bmatrix}, \quad \mathbf{u}_1 = \begin{bmatrix} -1 \\ 0 \\ 1 \end{bmatrix}, \quad \mathbf{u}_2 = \begin{bmatrix} 2 \\ 1 \\ 2 \end{bmatrix},$

Find the value for (i), (ii), and (iii).