每週問題 August 29, 2016

Let $A$ be an $m\times m$ matrix, $B$ be an $n\times m$ matrix and $C$ be an $n\times n$ matrix. If $A$ and $C$ are symmetric positive definite, show the following identities.
(a) $(A^{-1}+B^TC^{-1}B)^{-1}B^TC^{-1}=AB^T(BAB^T+C)^{-1}$
(b) $(A^{-1}+B^TC^{-1}B)^{-1}=A-AB^T(BAB^T+C)^{-1}BA$

每週問題 August 22, 2016

Let $\mathcal{V}$ and $\mathcal{W}$ be finite dimensional vector spaces, and $T:\mathcal{V}\to\mathcal{W}$ be a linear transformation. For a subspace $\mathcal{X}$ of $\mathcal{V}$, the image $T(\mathcal{X})=\{T(\mathbf{x})|\mathbf{x}\in\mathcal{X}\}$ of $\mathcal{X}$ under $T$ is a subspace of $\mathcal{W}$. Prove that if $\mathcal{X}\cap N(T)=\{\mathbf{0}\}$, then $\dim T(\mathcal{X})=\dim\mathcal{X}$. Note that $N(T)$ denotes the nullspace (kernel) of $T$.

每週問題 August 15, 2016

Let $S=\{\mathbf{0}\}$ be the set containing only the zero vector.
(a) Explain why $S$ must be linearly dependent.
(b) Explain why the empty set is a basis for $S$.

每週問題 August 8, 2016

For a set of vectors $S=\{\mathbf{v}_1,\ldots,\mathbf{v}_n\}$, prove that $\hbox{span}(S)$ is the intersection of all subspaces that contain $S$.

每週問題 August 1, 2016

For the matrix

$A=\begin{bmatrix} 1&0&0&1/3&1/3&1/3\\ 0&1&0&1/3&1/3&1/3\\ 0&0&1&1/3&1/3&1/3\\ 0&0&0&1/3&1/3&1/3\\ 0&0&0&1/3&1/3&1/3\\ 0&0&0&1/3&1/3&1/3 \end{bmatrix}$,

determine $A^{300}$.

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每週問題 July 25, 2016

Suppose that $A$ and $B$ are $m\times n$ complex matrices. If $A\mathbf{x}=B\mathbf{x}$ holds for every $\mathbf{x}\in\mathbb{C}^n$, prove that $A=B$.

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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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