## 一个向量组可由另一个向量组线性表示的充分必要条件是什么？（C019）

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## 选项

[A].   $r(A) > r(A,B)$

[B].   r(A) < r(A,B)

[C].   $r(A) = r(A,B)$

[D].   $r(A) \neq r(A,B)$

$r(A) = r(A,B)$

## 向量组的线性相关性与秩（C019）

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## 选项

[A].   $\mathrm{r}\left(\boldsymbol{\beta}_{1}, \boldsymbol{\beta}_{2}, \cdots, \boldsymbol{\beta}_{\mathrm{t}}\right)$ $=$ $\mathrm{r}\left(\boldsymbol{\alpha}_{1}, \boldsymbol{\alpha}_{2}, \cdots, \boldsymbol{\alpha}_{\mathrm{s}}\right)$

[B].   $\mathrm{r}\left(\boldsymbol{\beta}_{1}, \boldsymbol{\beta}_{2}, \cdots, \boldsymbol{\beta}_{\mathrm{t}}\right)$ $<$ $\mathrm{r}\left(\boldsymbol{\alpha}_{1}, \boldsymbol{\alpha}_{2}, \cdots, \boldsymbol{\alpha}_{\mathrm{s}}\right)$

[C].   $\mathrm{r}\left(\boldsymbol{\beta}_{1}, \boldsymbol{\beta}_{2}, \cdots, \boldsymbol{\beta}_{\mathrm{t}}\right)$ $\geqslant$ $\mathrm{r}\left(\boldsymbol{\alpha}_{1}, \boldsymbol{\alpha}_{2}, \cdots, \boldsymbol{\alpha}_{\mathrm{s}}\right)$

[D].   $\mathrm{r}\left(\boldsymbol{\beta}_{1}, \boldsymbol{\beta}_{2}, \cdots, \boldsymbol{\beta}_{\mathrm{t}}\right)$ $\leqslant$ $\mathrm{r}\left(\boldsymbol{\alpha}_{1}, \boldsymbol{\alpha}_{2}, \cdots, \boldsymbol{\alpha}_{\mathrm{s}}\right)$

$\mathrm{r}\left(\boldsymbol{\beta}_{1}, \boldsymbol{\beta}_{2}, \cdots, \boldsymbol{\beta}_{\mathrm{t}}\right)$ $\leqslant$ $\mathrm{r}\left(\boldsymbol{\alpha}_{1}, \boldsymbol{\alpha}_{2}, \cdots, \boldsymbol{\alpha}_{\mathrm{s}}\right)$

## 由向量的个数判断向量组的线性无关性（C019）

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## 选项

[A].   $t$ $>$ $s$

[B].   $t$ $=$ $s$

[C].   $t$ $\geqslant$ $s$

[D].   $t$ $\leqslant$ $s$

$t$ $\leqslant$ $s$

## 由向量的个数判断向量组的线性相关性（C019）

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## 选项

[A].   $\boldsymbol{\beta}_{1}$, $\boldsymbol{\beta}_{2}$, $\cdots$, $\boldsymbol{\beta}_{t}$ 一定是零向量组

[B].   $\boldsymbol{\beta}_{1}$, $\boldsymbol{\beta}_{2}$, $\cdots$, $\boldsymbol{\beta}_{t}$ 不存在

[C].   $\boldsymbol{\beta}_{1}$, $\boldsymbol{\beta}_{2}$, $\cdots$, $\boldsymbol{\beta}_{t}$ 线性无关

[D].   $\boldsymbol{\beta}_{1}$, $\boldsymbol{\beta}_{2}$, $\cdots$, $\boldsymbol{\beta}_{t}$ 线性相关

$\boldsymbol{\beta}_{1}$, $\boldsymbol{\beta}_{2}$, $\cdots$, $\boldsymbol{\beta}_{t}$ 线性相关

## 线性表示的部分与整体的关系（C019）

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## 选项

[A].   $\boldsymbol{\beta}$ 或许可由 $\boldsymbol{\alpha}_{1}$, $\boldsymbol{\alpha}_{2}$, $\cdots$, $\boldsymbol{\alpha}_{m}$ 线性表示

[B].   $\boldsymbol{\beta}$ 不可由 $\boldsymbol{\alpha}_{1}$, $\boldsymbol{\alpha}_{2}$, $\cdots$, $\boldsymbol{\alpha}_{m}$ 线性表示

[C].   $\boldsymbol{\beta}$ 可由 $\boldsymbol{\alpha}_{1}$, $\boldsymbol{\alpha}_{2}$, $\cdots$, $\boldsymbol{\alpha}_{m}$ 线性表示

[D].   $\boldsymbol{\beta}$ 可由 $\boldsymbol{\alpha}_{1}$, $\boldsymbol{\alpha}_{2}$, $\cdots$, $\boldsymbol{\alpha}_{m}$ 中的另一部分线性表示

$\boldsymbol{\beta}$ 可由 $\boldsymbol{\alpha}_{1}$, $\boldsymbol{\alpha}_{2}$, $\cdots$, $\boldsymbol{\alpha}_{m}$ 整体线性表示

## 线性相关与线性无关边缘处性质的推论（C019）

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## 选项

[A].   任一 $n$ 维向量 $\boldsymbol{\alpha}$ 不一定可由 $($ $\boldsymbol{\alpha}_{1}$, $\boldsymbol{\alpha}_{2}$, $\cdots$, $\boldsymbol{\alpha}_{n}$ $)$ 线性表示

[B].   任一 $n$ 维向量 $\boldsymbol{\alpha}$ 均不可由 $($ $\boldsymbol{\alpha}_{1}$, $\boldsymbol{\alpha}_{2}$, $\cdots$, $\boldsymbol{\alpha}_{n}$ $)$ 线性表示

[C].   任一 $n$ 维向量 $\boldsymbol{\alpha}$ 均可由 $($ $\boldsymbol{\alpha}_{1}$, $\boldsymbol{\alpha}_{2}$, $\cdots$, $\boldsymbol{\alpha}_{n}$ $)$ 线性表示，但表示法不唯一

[D].   任一 $n$ 维向量 $\boldsymbol{\alpha}$ 均可由 $($ $\boldsymbol{\alpha}_{1}$, $\boldsymbol{\alpha}_{2}$, $\cdots$, $\boldsymbol{\alpha}_{n}$ $)$ 线性表示，且表示法唯一

## 线性相关与线性无关边缘处的性质（C019）

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## 选项

[A].   $\boldsymbol{\beta}$ 不可由 $($ $\boldsymbol{\alpha}_{1}$, $\boldsymbol{\alpha}_{2}$, $\cdots$, $\boldsymbol{\alpha}_{\boldsymbol{r}}$ $)$ 线性表示

[B].   $\boldsymbol{\beta}$ 可由 $($ $\boldsymbol{\alpha}_{1}$, $\boldsymbol{\alpha}_{2}$, $\cdots$, $\boldsymbol{\alpha}_{\boldsymbol{r}}$ $)$ 线性表示，但表示法不唯一

[C].   $\boldsymbol{\beta}$ 可由 $($ $\boldsymbol{\alpha}_{1}$, $\boldsymbol{\alpha}_{2}$, $\cdots$, $\boldsymbol{\alpha}_{\boldsymbol{r}}$ $)$ 线性表示，且表示法唯一

[D].   $\boldsymbol{\beta}$ 或许可由 $($ $\boldsymbol{\alpha}_{1}$, $\boldsymbol{\alpha}_{2}$, $\cdots$, $\boldsymbol{\alpha}_{\boldsymbol{r}}$ $)$ 线性表示

$\boldsymbol{\beta}$ 可由 $($ $\boldsymbol{\alpha}_{1}$, $\boldsymbol{\alpha}_{2}$, $\cdots$, $\boldsymbol{\alpha}_{\boldsymbol{r}}$ $)$ 线性表示，且表示法唯一

## 选项

[A].   $\mathrm{r}\left(\boldsymbol{\alpha}_{1}, \boldsymbol{\alpha}_{2}, \cdots, \boldsymbol{\alpha}_{m} \right)$ $=$ $\mathrm{r}\left(\boldsymbol{\alpha}_{1}, \boldsymbol{\alpha}_{2}, \cdots, \boldsymbol{\alpha}_{m}, \boldsymbol{\beta}\right)$

[B].   $\mathrm{r}\left(\boldsymbol{\alpha}_{1}, \boldsymbol{\alpha}_{2}, \cdots, \boldsymbol{\alpha}_{m} \right)$ $\geqslant$ $\mathrm{r}\left(\boldsymbol{\alpha}_{1}, \boldsymbol{\alpha}_{2}, \cdots, \boldsymbol{\alpha}_{m}, \boldsymbol{\beta}\right)$

[C].   $\mathrm{r}\left(\boldsymbol{\alpha}_{1}, \boldsymbol{\alpha}_{2}, \cdots, \boldsymbol{\alpha}_{m} \right)$ $\leqslant$ $\mathrm{r}\left(\boldsymbol{\alpha}_{1}, \boldsymbol{\alpha}_{2}, \cdots, \boldsymbol{\alpha}_{m}, \boldsymbol{\beta}\right)$

[D].   $\mathrm{r}\left(\boldsymbol{\alpha}_{1}, \boldsymbol{\alpha}_{2}, \cdots, \boldsymbol{\alpha}_{m} \right)$ $\neq$ $\mathrm{r}\left(\boldsymbol{\alpha}_{1}, \boldsymbol{\alpha}_{2}, \cdots, \boldsymbol{\alpha}_{m}, \boldsymbol{\beta}\right)$

$\mathrm{\textcolor{red}{r}}\left(\boldsymbol{\alpha}_{1}, \boldsymbol{\alpha}_{2}, \cdots, \boldsymbol{\alpha}_{m} \right)$ $=$ $\mathrm{\textcolor{red}{r}}\left(\boldsymbol{\alpha}_{1}, \boldsymbol{\alpha}_{2}, \cdots, \boldsymbol{\alpha}_{m}, \boldsymbol{\textcolor{red}{\beta}}\right)$

## 选项

[A].   不确定

[B].

[C].   不能

$\textcolor{red}{\Leftrightarrow}$

## 选项

[A].

[B].   不能

[C].   不确定

$\textcolor{red}{\Leftrightarrow}$

## 选项

[A].   $\mathrm{r}\left(\boldsymbol{\alpha}_{1}, \boldsymbol{\alpha}_{2}, \cdots, \boldsymbol{\alpha}_{m} \right)$ $\geqslant$ $m$

[B].   $\mathrm{r}\left(\boldsymbol{\alpha}_{1}, \boldsymbol{\alpha}_{2}, \cdots, \boldsymbol{\alpha}_{m} \right)$ $\leqslant$ $m$

[C].   $\mathrm{r}\left(\boldsymbol{\alpha}_{1}, \boldsymbol{\alpha}_{2}, \cdots, \boldsymbol{\alpha}_{m} \right)$ $=$ $m$

[D].   $\mathrm{r}\left(\boldsymbol{\alpha}_{1}, \boldsymbol{\alpha}_{2}, \cdots, \boldsymbol{\alpha}_{m} \right)$ $<$ $m$

$\textcolor{yellow}{\Leftrightarrow}$
$\textcolor{orange}{\mathrm{r}}\left(\textcolor{cyan}{\boldsymbol{\alpha}_{1}}, \textcolor{cyan}{\boldsymbol{\alpha}_{2}}, \textcolor{cyan}{\cdots}, \textcolor{cyan}{\boldsymbol{\alpha}_{m}} \right)$ $<$ $\textcolor{red}{m}$

## 选项

[A].   $\mathrm{r}\left(\boldsymbol{\alpha}_{1}, \boldsymbol{\alpha}_{2}, \cdots, \boldsymbol{\alpha}_{m} \right)$ $\leqslant$ $m$

[B].   $\mathrm{r}\left(\boldsymbol{\alpha}_{1}, \boldsymbol{\alpha}_{2}, \cdots, \boldsymbol{\alpha}_{m} \right)$ $<$ $m$

[C].   $\mathrm{r}\left(\boldsymbol{\alpha}_{1}, \boldsymbol{\alpha}_{2}, \cdots, \boldsymbol{\alpha}_{m} \right)$ $=$ $m$

[D].   $\mathrm{r}\left(\boldsymbol{\alpha}_{1}, \boldsymbol{\alpha}_{2}, \cdots, \boldsymbol{\alpha}_{m} \right)$ $\geqslant$ $m$

$\textcolor{yellow}{\Leftrightarrow}$
$\textcolor{orange}{\mathrm{r}}\left(\textcolor{cyan}{\boldsymbol{\alpha}_{1}}, \textcolor{cyan}{\boldsymbol{\alpha}_{2}}, \textcolor{cyan}{\cdots}, \textcolor{cyan}{\boldsymbol{\alpha}_{m}} \right)$ $=$ $\textcolor{red}{m}$

## 选项

[A].   向量组中向量的个数

[B].   极大无关组中所含向量的维度

[C].   极大无关组中所含向量的个数

[D].   向量组中非零向量的个数