DSQ 向量空間1

(1) Let $S$ be the set of all real ordered pairs. Is $S$ a vector space?

1. $(x_1,y_1)+(x_2,y_2)=(x_1+x_2,y_1+y_2)$ for all $(x_1,y_1), (x_2,y_2)\in S$.
2. $\alpha(x,y)=(\alpha x,0)$ for all $(x,y)\in S$ and $\alpha\in\mathbb{R}$.

(2) Let $\mathcal{M}_{n}(\mathbb{R})$ denote the vector space formed by all $n\times n$ real matrices. Is $S$ a subspace of $\mathcal{M}_n(\mathbb{R})$?

1. $S$ is the set of all upper triangular matrices.
2. $S$ is the set of all matrices with zero trace.

(3) Let $\mathbf{u}_1,\mathbf{u}_2,\mathbf{u}_3$ be vectors in $\mathbb{R}^3$. What is the dimension of $\text{span}\{\mathbf{u}_1,\mathbf{u}_2,\mathbf{u}_3\}$?

1. $\mathbf{u}_1+\mathbf{u}_2=(0,0,0)$
2. $\mathbf{u}_2+\mathbf{u}_3=(0,0,0)$

(4) Let $W=\left\{\left.\begin{bmatrix} a&b\\ c&d \end{bmatrix}\right\vert a,b,c,d\in\mathbb{R}\right\}$. What is the dimension of $W$?

1. For any $A$, $AB=BA$ for all $B\in M$.
2. $abcd=0$

(5) Is $\mathbf{v}_3$ a linear combination of $\mathbf{v}_1$ and $\mathbf{v}_2$?

1. $\mathbf{v}_1,\mathbf{v}_2,\mathbf{v}_3$ are linearly dependent.
2. $\mathbf{v}_2,\mathbf{v}_3,\mathbf{v}_4$ are linearly independent.

(6) Let $\mathbf{v}_1,\ldots,\mathbf{v}_n$ be vectors in vector space $\mathcal{V}$. Is $\boldsymbol{\beta}=\{\mathbf{v}_1,\ldots,\mathbf{v}_n\}$ a basis for $\mathcal{V}$?

1. The vector space $\mathcal{V}$ can be spanned by $\mathbf{v}_1,\ldots,\mathbf{v}_n$, and $\boldsymbol{\beta}$ is linearly independent.
2. Every $\mathbf{u}\in\mathcal{V}$ can be expressed as a linear combination of $\mathbf{v}_1,\ldots,\mathbf{v}_n$, and any vector in $\boldsymbol{\beta}$ is not a linear combination of the remaining vectors.

(7) Let $S$ be a set of $k$ vectors in $\mathbb{R}^n$. Is $k?

1. $\mathbb{R}^n\neq\text{span}(S)$
2. The vectors in $S$ are linearly independent.

(8) Are $\mathbf{v}_1,\mathbf{v}_2,\mathbf{v}_3$ linearly dependent?

1. $\mathbf{v}_1+\mathbf{v}_2,\mathbf{v}_2+\mathbf{v}_3,\mathbf{v}_3+\mathbf{v}_1$ are linearly dependent.
2. $\mathbf{v}_1-\mathbf{v}_2,\mathbf{v}_2-\mathbf{v}_3,\mathbf{v}_3-\mathbf{v}_1$ are linearly dependent.

(9) What is the dimension of the span of $A\mathbf{v}_1, A\mathbf{v}_2, A\mathbf{v}_3$?

1. $\mathbf{v}_1,\mathbf{v}_2,\mathbf{v}_3$ are linearly independent.
2. The columns of $A$ are linearly independent.

(10) Let $\mathcal{X}$ and $\mathcal{Y}$ be two nontrivial subspaces in $\mathbb{R}^6$. The sum of $\mathcal{X}$ and $\mathcal{Y}$ is defined by $\mathcal{X}+\mathcal{Y}=\{\mathbf{x}+\mathbf{y}\vert\mathbf{x}\in\mathcal{X},\mathbf{y}\in\mathcal{Y}\}$. What is the dimension of $\mathcal{X}+\mathcal{Y}$?

1. $\dim\mathcal{Y}=\dim\mathcal{X}+2$
2. $\mathcal{X}\cap\mathcal{Y}=\{\mathbf{0}\}$

(1) B

(2) D
$\alpha$$\beta$ 是任何數。若 $A$$B$ 為上三角矩陣，則 $\alpha A+\beta B$ 亦為上三角矩陣。因此，所有的上三角矩陣形成的集合為一子空間。若 $\text{trace}A=0$$\text{trace}B=0$，利用跡數的線性函數性質，可得 $\text{trace}(\alpha A+\beta B)=\alpha\text{trace}A+\beta\text{trace}B=0$。因此，所有跡數等於零的矩陣所形成的集合為一子空間。

(3) E
$A=\begin{bmatrix} \mathbf{u}_1&\mathbf{u}_2&\mathbf{u}_3 \end{bmatrix}$。陳述 (1) 說明 $(1,1,0)$ 屬於零空間 $\ker A$，陳述 (2) 說明 $(0,1,1)$ 屬於 $\ker A$，縱使合併二個陳述只能確認 $\dim\ker A\ge 2$。根據秩─零度定理，$\text{span}\{\mathbf{u}_1,\mathbf{u}_2,\mathbf{u}_3\}$ 的維數，即 $A$ 的秩為 $\text{rank}A=3-\dim\ker A$，也就是說 $\text{rank}A=0$$1$

(4) A

(5) C

(6) D

(7) C

(8) A
$\mathbf{v}_1+\mathbf{v}_2,\mathbf{v}_2+\mathbf{v}_3,\mathbf{v}_3+\mathbf{v}_1$ 為一線性相關集，則存在不全為零的數組 $c_1,c_2,c_3$ 使得

$c_1(\mathbf{v}_1+\mathbf{v}_2)+c_2(\mathbf{v}_2+\mathbf{v}_3)+c_3(\mathbf{v}_3+\mathbf{v}_1)= (c_1+c_3)\mathbf{v}_1+(c_1+c_2)\mathbf{v}_2+(c_2+c_3)\mathbf{v}_3=\mathbf{0}$

$\begin{bmatrix} 1&0&1\\ 1&1&0\\ 0&1&1 \end{bmatrix}\begin{bmatrix} c_1\\ c_2\\ c_3 \end{bmatrix}=\begin{bmatrix} 0\\ 0\\ 0 \end{bmatrix}$

(9) C
$A$ 為一 $m\times n$ 階矩陣，則 $\mathbf{v}_i$$n$ 維向量。若 $\mathbf{v}_1,\mathbf{v}_2,\mathbf{v}_3$ 組成線性獨立集，則 $\dim\text{span}\{A\mathbf{v}_1,A\mathbf{v}_2,A\mathbf{v}_3\}\le 3$。若 $A$ 有線性獨立的行向量，則 $\ker A=\{\mathbf{0}\}$。如果二個條件同時成立，考慮

$c_1A\mathbf{v}_1+c_2A\mathbf{v}_2+c_3A\mathbf{v}_3=A(c_1\mathbf{v}_1+c_2\mathbf{v}_2+c_3\mathbf{v}_3)=\mathbf{0}$

(10) E