101年 - 101 國立嘉義大學_碩士班(甲組)招生考試試題_應用數學系：線性代數#146244

1. Compute the symmetric $LDL^T$ factorization of $\begin{pmatrix} a & b \\ c & d \end{pmatrix}$. (10%)

2. Find the determinant of $A = (a_{i,j})$, if $a_{i,j} = i + j$ for $i, j \in \{1, 2, \cdots, n\}$. (10%)

3. Let A be an $m \times n$ matrix over $\mathbb{R}$, prove that $rank(A) \leq m$ and $rank(A) \leq n$. (10%)

4. Prove that $A = \begin{pmatrix} 2 & 1 & 0 \\ 0 & 3 & 1 \\ 0 & 0 & 4 \end{pmatrix}$ and $B = \begin{pmatrix} 2 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 4 \end{pmatrix}$ are similar. (10%)

(1) Suppose that $\{v_1, v_2\}$ is an orthonormal basis for $V$ and $T: V \to V$ is a self-adjoint linear map. Show that the matrix of $T$ relative to that basis is symmetric. (15%)

(2) Assume that $f: V \times V \to R$ is a map defined by $f(v, w) = \langle T(v), w \rangle$ where $T$ is a self-adjoint linear map. Show that $f$ is a bilinear symmetric form of $V$. (15%)

6. Let $A = \begin{pmatrix} 1 & 1 \\ 1 & 2 \\ 1 & 4 \end{pmatrix}, y = \begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix}$. Find $\min_{x \in \mathbb{R}^2} \|Ax - y\|$. (15%)

7. In $\mathbb{R}^4$, let $w_1 = (1, 0, 1, 0)$, $w_2 = (0, 1, 0, 1)$, $w_3 = (1, 0, 0, 0)$. Prove that $\{w_1, w_2, w_3\}$ is linearly independent. Use the Gram-Schmidt process to compute the orthogonal vectors $v_1, v_2$ and $v_3$ such that $span\{w_1, w_2, w_3\} = span\{v_1, v_2, v_3\}$ and then normalize these vectors to obtain an orthonormal set. (15%)