## DSQ 向量空間2

(1) Let $C(A)$ be the column space of $A$ and $N(A)$ be the nullspace of $A$. Is $A=\begin{bmatrix} a&b\\ c&d \end{bmatrix}$ the zero matrix?

1. $\dim C(A)<2$
2. $\dim C(A)<\dim N(A)$

(2) If $a,b,c,d$ are real, what is the rank of $A=\begin{bmatrix} a&b\\ c&d \end{bmatrix}$?

1. $a+b=c+d=0$
2. $c=1$

(3) Let $A$ be an $m\times n$ real matrix. What is the row space of $A$?

1. $A$ has linearly independent columns.
2. The nullspace of $A$ contains only the zero vector.

(4) Let $A$ be an $m\times n$ real matrix. Is $A\mathbf{x}=\mathbf{b}$ consistent (solvable) for all $\mathbf{b}\in\mathbb{R}^m$?

1. $A$ has linearly independent rows.
2. The nullspace of $A$ contains only the zero vector.

(5) For $a,b,c\in\mathbb{R}$, what is $A=\begin{bmatrix} a&b&c \end{bmatrix}$?

1. The nullspace of $A$ consists of all vectors $(x,y,z)$ such that $x+2y+3z=0$.
2. $a=5$

(6) If $A=\begin{bmatrix} 1&b\\ c&d \end{bmatrix}$, what is the value of $d$?

1. $A$ has rank one and $b=2$.
2. $A$ has rank one and $c=0$.

(7) Let $B=\begin{bmatrix} A&\mathbf{b} \end{bmatrix}$, where $A$ is an $m\times n$ real matrix and $\mathbf{b}$ is an $m$-dimensional real vector. Does $A\mathbf{x}=\mathbf{b}$ has a unique solution?

1. $\text{rank}B=\text{rank}A=n$
2. $\text{rank}B=\text{rank}A=m$

(8) Let $A$ and $B$ be $n\times n$ matrices. Is $\text{rank}(A^2)=\text{rank}(B^2)$?

1. $\text{rank}A=\text{rank}B$
2. $AB=0$

(9) Let $A$ and $B$ be $n\times n$ matrices. Is $A=B$?

1. The column space of $A$ equals the column space of $B$ and the nullspace of $A$ equals the nullspace of $B$.
2. The column space of $A^T$ equals the column space of $B^T$ and the nullspace of $A^T$ equals the nullspace of $B^T$.

(10) If $A$ is an $m\times n$ matrix of rank $r$, is $r?

1. There are vectors $\mathbf{b}$ for which $A\mathbf{x}=\mathbf{b}$ has no solution.
2. There are vectors $\mathbf{b}$ for which $A\mathbf{x}=\mathbf{b}$ has infinitely many solutions.

(1) B

(2) C
$a+b=0$$c+d=0$，則 $(1,1)$ 屬於零空間 $N(A)$，可知 $\dim N(A)>0$。若 $c=1$，則行空間 $C(A)$ 包含非零向量，即 $\dim C(A)>0$。使用秩─零度定理 $\dim C(A)+\dim N(A)=2$，合併陳述 (1) 和 (2) 可得 $\dim C(A)=1$$\dim N(A)=1$，所以 $\text{rank}A=\dim C(A)=1$

(3) D
$A$ 有線性獨立的行向量，則 $\text{rank}A=\dim C(A)=\dim C(A^T)=n$，即知 $C(A^T)=\mathbb{R}^n$。若 $N(A)=\{\mathbf{0}\}$，使用補子空間性質 $C(A^T)\cap N(A)=\{\mathbf{0}\}$$C(A^T)+N(A)=\mathbb{R}^n$，可得 $C(A^T)=\mathbb{R}^n$。以上結果顯示陳述 (1) 等價於陳述 (2)。

(4) A
$A$ 有線性獨立的列向量，則 $\text{rank}A=\dim C(A)=m$，表示行空間 $C(A)$ 充滿整個 $\mathbb{R}^m$，所以 $A\mathbf{x}=\mathbf{b}$ 必定有解。若 $N(A)=\{\mathbf{0}\}$，則 $C(A^T)=\mathbb{R}^n$。當 $m>n$，行空間 $C(A)$ 未能充滿整個 $\mathbb{R}^m$，故不保證 $A\mathbf{x}=\mathbf{b}$ 總是存在一解。

(5) C

(6) B
$\text{rank}A=1$。若 $b=2$，則 $(1,2)$$(c,d)$ 線性相關，故 $d=2c$。但 $c$ 未給定，因此無法確定 $d$。若 $c=0$，則 $(1,b)$$(0,d)$ 線性相關，推論 $d=0$

(7) A

(8) E

(9) E

(10) B

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