103年 - 103 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)：基礎數學 (含微積分、高等微積 分及線性代數)#147995

科目：【阿摩】未分類題庫 | 年份：103年 | 選擇題數：0 | 申論題數：12

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所屬科目：【阿摩】未分類題庫

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1 a. (5) State the Fundamental Theorem of Calculus.
b. (5) If $f$ is a continuous function such that

$$\int_{1}^{x} f(t)dt = (x - 1)e^{2x} + \int_{1}^{x} e^{-t}f(t)dt$$

for all $x$, find $f(x)$.
2 Let $U_k$ be a sequence of open bounded sets in $\mathbb{R}^n$. Prove or disprove:

a. (5) $\bigcup_{k=1}^{\infty} U_k$ is open.
b. (5) $\bigcap_{k=1}^{\infty} U_k$ is open.
3 (10) Suppose that $f : [0, 1] \to \mathbb{R}$ is continuous function. Show that $f$ is uniformly continuous on $[0, 1]$.
4 a. (10) Let $E \subset \mathbb{R}$ and suppose that $f_n \to f$ uniformly on $E$, as $n \to \infty$. If each $f_n$ is continuous, then $f$ is continuous.

b. (10) Let $f_n(x) = x^n$, $0 \leqslant x \leqslant 1$. Does $f_n$ converge uniformly?. What if $0 \leqslant x < 1$?
5 Suppose that $A \in M_{n \times n}(\mathbb{R})$. Let $T : M_{n \times n}(\mathbb{R}) \to M_{n \times n}(\mathbb{R})$ be defined by $T(X) = AX - XA$.

a. (5) Show that $T$ is a linear transformation.
b. (10) Let $A = \begin{bmatrix} 3 & 2 \\ 2 & 0 \end{bmatrix}$. Find the eigenvalues and eigenvectors of $T$.
6 (10) Suppose that $A \in M_{m \times n}(\mathbb{R})$ and $\operatorname{rank}(A) = n$. Show that $A^T A \in M_{n \times n}(\mathbb{R})$ is an invertible matrix.
7 (10) Let $A, B \in M_{n \times n}(\mathbb{R})$. If $B$ is invertible, then show that there exist a scalar $\lambda \in \mathbb{C}$ and a nonzero vector $x \in \mathbb{C}^n$ such that $Ax = \lambda Bx$.
8 Let $\| \cdot \| : M_{2 \times 2}(\mathbb{R}) \to \mathbb{R}$ be a function defined by $\| A \| = \max_{x \in S} \| Ax \|_2$, where $S = \{x \in \mathbb{R}^2 | \| x \|_2 = 1\}$.

a. (5) Let $A = \begin{bmatrix} 3 & 2 \\ 2 & 0 \end{bmatrix}$. Find $\| A \|$.
b. (10) Show that $(M_{2 \times 2}(\mathbb{R}), \| \cdot \|)$ is a normed space.