102年 - 102 國立高雄大學_博士班招生考試試題_應用數學系(一般生、在職生)：線性代數#147992

科目：研究所、轉學考(插大)◆線性代數 | 年份：102年 | 選擇題數：0 | 申論題數：11

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所屬科目：研究所、轉學考(插大)◆線性代數

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申論題 (11)

1 (10) Let $T: \mathbb{R}^n \to \mathbb{R}^m$ be a linear transformation. Show that

$$n = \operatorname{rank}(T) + \operatorname{nullity}(T).$$
2 Let $A, B \in M_{n \times n}(\mathbb{R})$.

a. (10) Show that \(\operatorname{rank}(AB) \leqslant \operatorname{rank}(B)\) and \(\operatorname{rank}(AB) \leqslant \operatorname{rank}(A)\).
b. (10) Show that \(\operatorname{rank}(A^{\top}A) = \operatorname{rank}(A)\).
c. (5) If \( A \) is invertible then show that \( \operatorname{rank}(AB) = \operatorname{rank}(B) \).
3 Let $A \in M_{n \times n}(\mathbb{R})$ be a symmetric matrix

a. (10) Show that the eigenvalues of \( A \) are real.
b. (10) If \(\lambda{1}\) and \(\lambda{2}\) are distinct eigenvalues of \(A\) associated eigenvectors \(x{1}\) and \(x{2}\), then \(x{1}\) and \(x{2}\) are orthogonal.
4 Suppose that $A \in M{n \times k}(\mathbb{R})$ and $B \in M{m \times \ell}(\mathbb{R})$. Let $T: M{k \times m}(\mathbb{R}) \to M{n \times \ell}(\mathbb{R})$ be defined by $T(X) = AXB$.

a. (5) Show that \( T \) is a linear transformation.
b. (10) Let \( A = \left[ \begin{array}{ll}3 & 2\\ 2 & 0 \end{array} \right] \) and \( B = I_{2} \). Find the eigenvalues and eigenvectors of \( T \).
5 Let $A = \left[ \begin{array}{cccc} 1 & 0 & 1 & 1 \\ 1 & 1 & 0 & 1 \end{array} \right]^\top \in M_{4 \times 2}(\mathbb{R})$ and $b = [1, 1, 0, 0]^\top \in \mathbb{R}^4$.

a. (10) Find \(\min{x\in S}\| Ax\| 2\) and \(\max{x\in S}\| Ax\| 2\) , where \(S = \{x\in \mathbb{R}^2\mid \| x\| _2 = 1\}\)
b. (10) Find \(x0 \in \mathbb{R}^2\) such that \(\|Ax0 - b\|2 = \min{x \in \mathbb{R}^2} \|Ax - b\|_2\).
6 (10) Let $A, B$ and $D$ be real matrices. Suppose that $A = A^\top$, $D = D^\top$ and $A$ is invertible. Then show that $\left[ \begin{array}{cc} A & B \\ B^\top & D \end{array} \right]$ is positive if and only if $A$ and $D - B^\top A^{-1}B$ are positive definite.