## 每週問題 July 18, 2016

Prove that each of the following statements is true.
(a) If $A=[a_{ij}]$ is skew symmetric, then $a_{ii}=0$ for each $i$.
(b) If $A=[a_{ij}]$ is skew Hermitian, then each $a_{ii}$ is a pure imaginary number.
(c) If $A$ is real and symmetric, then $B=\mathrm{i}A$ is skew Hermitian, where $\mathrm{i}=\sqrt{-1}$.

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## 每週問題 July 11, 2016

Suppose that $A$ is the coefficient matrix for a homogeneous system of four equations in six unknowns and suppose that $A$ has at least one nonzero row.
(a) Determine the fewest number of free variables that are possible.
(b) Determine the maximum number of free variables that are possible.

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## 每週問題 July 4, 2016

Suppose that $\begin{bmatrix} A|\mathbf{b}\end{bmatrix}$ is reduced to a matrix $\begin{bmatrix} E|\mathbf{c} \end{bmatrix}$.
(a) Is $\begin{bmatrix} E|\mathbf{c} \end{bmatrix}$ in row echelon form if $E$ is?
(b) If $\begin{bmatrix} E|\mathbf{c} \end{bmatrix}$ is in row echelon form, must $E$ be in row echelon form?

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## 每週問題 June 27, 2016

Let $A$ be an $n\times n$ matrix. If $A\mathbf{x}=\mathbf{0}$ has nonzero solutions, is it possible that $A^T\mathbf{x}=\mathbf{b}$ has a unique solution for some vector $\mathbf{b}$?

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## 每週問題 June 20, 2016

Let $\{\mathbf{q}_1,\mathbf{q}_2,\mathbf{q}_3\}$ be an orthonormal set in $\mathbb{R}^3$ and $\mathbf{q}_3$ be the cross product of $\mathbf{q}_1$ and $\mathbf{q}_2$, i.e., $\mathbf{q}_3=\mathbf{q}_1\times\mathbf{q}_2$. A linear transformation $T:\mathbb{R}^3\to\mathbb{R}^3$ is defined by

$T(\mathbf{x})=\mathbf{x}\times \mathbf{q}_1+(\mathbf{q}_2^T\mathbf{x})\mathbf{q}_1$.

Determine the rank of $T$.

## 每週問題 June 13, 2016

Let $\boldsymbol{\beta}=\{\mathbf{x}_1,\ldots,\mathbf{x}_k\}$ and $\boldsymbol{\gamma}=\{\mathbf{y}_1,\ldots,\mathbf{y}_k\}$ be bases for a subspace $\mathcal{V}$ in $\mathbb{R}^n$. Let $X=\begin{bmatrix} \mathbf{x}_1&\cdots&\mathbf{x}_k \end{bmatrix}$ and $Y=\begin{bmatrix} \mathbf{y}_1&\cdots&\mathbf{y}_k \end{bmatrix}$. Show that the change of coordinates matrix from $\boldsymbol{\beta}$ to $\boldsymbol{\gamma}$ is

$P=(Y^TY)^{-1}Y^TX$.

## 每週問題 June 6, 2016

Let $A$ be an $m\times n$ complex matrix. If $A\mathbf{x}=\mathbf{b}$ is consistent for some $\mathbf{b}$, prove that there exists a unique solution $\mathbf{x}$ in the column space of $A^\ast$.

## 每週問題 May 30, 2016

Let $A$ be an $m\times n$ matrix and $S$ be the solution set for a consistent system of linear equations $A\mathbf{x}=\mathbf{b}$ for some $\mathbf{b}\neq\mathbf{0}$.
(a) If $S_{\max}$ is a maximal independent subset of $S$ and $\mathbf{x}_p$ is any particular solution, show that

$\hbox{span}(S_{\max})=\hbox{span}\{\mathbf{x}_p\}+N(A)$,

where $N(A)$ denotes the nullspace of $A$.
(b) If $\hbox{rank}A=r$, show that $A\mathbf{x}=\mathbf{b}$ has at most $n-r+1$ independent solutions.

## 每週問題 May 23, 2016

Let $A$ and $B$ be $m\times n$ matrices. If $\hbox{rank}A=r$ and $\hbox{rank}B=k\le r$, show that

$r-k\le \hbox{rank}(A+B)\le r+k$.

In words, a perturbation of rank $k$ can change the rank by at most $k$.

## Cayley-Hamilton 定理的一個錯誤「證明」

\begin{aligned} p(\lambda)&=\det(A-\lambda I)=\begin{vmatrix} a-\lambda&b\\ c&d-\lambda \end{vmatrix}\\ &=\lambda^2-(a+d)\lambda+ad-bc,\end{aligned}

Cayley-Hamilton 定理宣稱

$p(A)=A^2-(a+d)A+(ad-bc)I=0$

Cayley-Hamilton 定理有很多種證法 (見“Cayley-Hamilton 定理”)，但其中幾乎挑不出一個簡單的證明。底下這個看似快捷實乃錯誤的「證明」曾經不斷地被初學者重複發現：

$p(A)=\det(A-AI)=\det(A-A)=\det 0=0$