113年 - 113 國立臺北大學_碩士班一般入學考試試題_統計學系：基礎數學#147042

1. Let $A = [a_{ij}]$ be an $m \times n$ matrices, $B = [b_{ij}]$ be an $m \times m$ matrices and $\lambda$ be a given constant. Use Sigma notation (i.e. $\Sigma$) and $a_{ij}, b_{ij}$ to represent $\text{trace}(\lambda I - A^T BA)$. (10%)

2. Let $A = \begin{bmatrix} 1 & -1 & 2 & 1 & 1 \\ 1 & -1 & 2 & 2 & 1 \\ 1 & -1 & 4 & 1 & 1 \\ 1 & -2 & 2 & 1 & 1 \\ -2 & 2 & -4 & -2 & 1 \end{bmatrix}$. Find the inverse of A. (10%)

(a) Prove that \( T \) is linear. \( (7\%) \)

(b) Let \( B = \{1 + 2x, 1 - x\} \) and \( C = \{1, x, x^2 + 1\} \). Determine \( [T]_{C \leftarrow B} \), i.e. the matrix of \( T \) with respect to bases \( B \) and \( C \). (7%)

(c) Determine the rank and nullity of \( T \). (6%)

4. Let $S = \{\vec{v}_1, \vec{v}_2, \dots, \vec{v}_n\}$ be a basis for $R^n$ and $A$ be a $n \times n$ nonsingular matrix. Please show that $\{A\vec{v}_1, A\vec{v}_2, \dots, A\vec{v}_n\}$ is linear independent. (10%) 

第二大題(共 50%)
1. Calculate the \( \int_{-\infty}^{\infty}e^{-\frac{x^2}{2}}dx. \) (15%)

2. Let the \( f(\theta) = \frac{n!}{(n - x)!x!}\theta^x (1 - \theta)^{n - x}, x < n \), be the function of \( \theta \in [0,1] \).
Denoted $F(\theta) = \log f(\theta)$, where x and n are the positive, fixed constants. Show that x/n is the unique critical point that is a maximizer of F concerning $\theta$. (15%)

3. Find the \( \lim_{x\to +\infty}\frac{e - (1 + x)^{\frac{1}{2}}}{x}. \) (10%)

4. Calculate the \( \int_0^1 (4x^3 -3x^2)dx \) by definition of Riemann sum.
Hint: $\sum_{i=1}^n i^2 = \frac{n(n+1)(2n+1)}{6}$ and $\sum_{i=1}^n i^3 = \left[\frac{n(n+1)}{2}\right]^2$. (10%)