## 每週問題 August 11, 2014

Consider an $m\times n$ real matrix $A$ with linearly independent columns, and $m>n$. Which of the following statements are true?

(a) $A^TA$ is positive definite.
(b) $AA^T$ is positive definite.
(c) The column space of $A$ is spanned by all the eigenvectors of $AA^T$.
(d) The row space of $A$ is spanned by all the eigenvectors of $A^TA$.
(e) $A$ and $AA^TA$ have the same column space.

## 每週問題 August 4, 2014

Let

$\displaystyle A=\frac{1}{2}\left[\!\!\begin{array}{crrr} 1&1&\sqrt{2}&0\\ 1&1&-\sqrt{2}&0\\ 1&-1&0&\sqrt{2}\\ 1&-1&0&-\sqrt{2} \end{array}\!\!\right]\left[\!\!\begin{array}{ccr} 1&1&0\\ 0&2&0\\ 0&0&-2\\ 0&0&0 \end{array}\!\!\right]\left[\!\!\begin{array}{rrc} \frac{1}{\sqrt{2}}&\frac{1}{\sqrt{6}}&\frac{1}{\sqrt{3}}\\[0.5em] -\frac{1}{\sqrt{2}}&\frac{1}{\sqrt{6}}&\frac{1}{\sqrt{3}}\\[0.5em] 0&-\frac{2}{\sqrt{6}}&\frac{1}{\sqrt{3}}\end{array}\!\!\right]$.

Find $\text{rank}\begin{bmatrix} AA^T&A\\ A^T&A^TA \end{bmatrix}$.

## 每週問題 July 28, 2014

Consider a $4\times 4$ real matrix $A$ with three different eigenvalues $0,1,2$. Which of the following statements are true?

(a) The determinant of $A$ is $0$.
(b) There are three linearly independent eigenvectors.
(c) The rank of $A$ is $2$.
(d) The trace of $A$ is $3$.
(e) The column space of $A$ is spanned by the eigenvectors corresponding to eigenvalues $1$ and $2$.

## 每週問題 July 21, 2014

Let $f(t)$ be defined as follows:

$\displaystyle f(t)=\left\{\begin{array}{rl} 1,&\text{if~}0\le t<\pi\\ -1,&\text{if~}-\pi

Also, define

$\displaystyle g(t)=a\cos t+b\cos 2t+c\sin t.$

Find the coeffiicents $(a,b,c)$ such that $E=\int_{-\pi}^{\pi}\vert g(t)-f(t)\vert^2 dt$ is minimized.

## 每週問題 July 14, 2014

Let $A$ be an $n\times n$ real matrix. Which of the following statements are true?

(a) If all the eigenvalues of $A$ are positive, then $\mathbf{x}^TA\mathbf{x}>0$ for every nonzero $\mathbf{x}\in\mathbb{R}^n$.
(b) If all the eigenvalues of $A$ are positive, then $\det(A+A^T)>0$.
(c) If $\mathbf{x}^TA\mathbf{x}>0$ for every nonzero $\mathbf{x}\in\mathbb{R}^n$, then $\det A>0$.
(d) If $\mathbf{x}^TA\mathbf{x}<0$ for every nonzero $\mathbf{x}\in\mathbb{R}^n$, then $\det A<0$.
(e) If $\mathbf{x}^TA\mathbf{x}>0$ for every nonzero $\mathbf{x}\in\mathbb{R}^n$, then $\det (A+A^T)>0$.

## 每週問題 July 7, 2014

Let

$\displaystyle A=\left[\!\!\begin{array}{cr} 1&-1\\ 1&0 \end{array}\!\!\right]$.

Find the minimum positive integer $n$ such that $A^n=I$.

## 2014 年大學指考數甲的線性代數問題

(1) 若 $(x,y,z)$ 為此方程組的解，則 $x=0$

(2) 若 $(x,y,z)$ 為此方程組的解，則 $y>0$

(3) 若 $(x,y,z)$ 為此方程組的解，則 $y

(4) 當 $a\neq -3$ 時，恰有一組 $(x,y,z)$ 滿足此方程組

(5) 當 $a=-3$ 時，$(x,y,z)$ 滿足此方程組的所有解 $(x,y,z)$ 會在一條直線上

## 每週問題 June 30, 2014

Let

$\displaystyle A=\left[\!\!\begin{array}{rcc} 3&3&3\\ -6&6&2\\ 7&1&2 \end{array}\!\!\right]\left[\!\!\begin{array}{ccr} 3&0&0\\ 0&0&0\\ 0&0&-2 \end{array}\!\!\right]\left[\!\!\begin{array}{rcc} 3&3&3\\ -6&6&2\\ 7&1&2 \end{array}\!\!\right]^{-1}$

and $\mathbf{b}=\left[\!\!\begin{array}{r} 15\\ -14\\ 25 \end{array}\!\!\right]$.

(a) Find the general solution (also called the complete solution) of $A\mathbf{x}=\mathbf{b}$.
(b) Find the distance from $\mathbf{b}$ to the row space of $A$.

## 共形映射

$A$ 為一 $n\times n$ 階實矩陣。我們可以將 $A$ 視為一個從幾何向量空間 $\mathbb{R}^n$ 映至 $\mathbb{R}^n$ 的線性變換：$\mathbf{x}\mapsto A\mathbf{x}$，其中 $\mathbf{x}\in\mathbb{R}^n$。如果線性變換 $A$ 不改變向量長度，則 $A$ 稱為保長 (length-preserving) 映射或等距同構 (isometry)。保長映射 $A$ 有下列等價的定義方式 (見“等距同構與么正矩陣”)：

1. $A$ 是一正交 (orthogonal) 矩陣，即 $A^TA=AA^T=I$
2. 對於每一 $\mathbf{x}\in\mathbb{R}^n$$\Vert A\mathbf{x}\Vert=\Vert\mathbf{x}\Vert$
3. 對於任意 $\mathbf{x},\mathbf{y}\in\mathbb{R}^n$$\Vert A\mathbf{x}-A\mathbf{y}\Vert=\Vert\mathbf{x}-\mathbf{y}\Vert$
4. 對於任意 $\mathbf{x},\mathbf{y}\in\mathbb{R}^n$$(A\mathbf{x})^T(A\mathbf{y})=\mathbf{x}^T\mathbf{y}$

$\displaystyle \mathbf{x}^T\mathbf{y}=\Vert\mathbf{x}\Vert \Vert\mathbf{y}\Vert\cos\theta$

$\displaystyle \frac{(A\mathbf{x})^T(A\mathbf{y})}{\Vert A\mathbf{x}\Vert\Vert A\mathbf{y}\Vert}=\frac{\mathbf{x}^T\mathbf{y}}{\Vert \mathbf{x}\Vert\Vert \mathbf{y}\Vert}$

$A$ 稱為保角 (angle-preserving) 映射。這個定義隱含了 $A$ 必須是一個可逆矩陣，否則存在 $\mathbf{x}\neq\mathbf{0}$ 使得 $A\mathbf{x}=\mathbf{0}$，如此便無從計算夾角。