向量可由向量组线性表示的充要条件:所形成的矩阵的秩(C019)

问题

以下哪个选项可以说明向量 $\textcolor{orange}{\boldsymbol{\beta}}$ 和向量组 $\textcolor{yellow}{\boldsymbol{\alpha}_{1}}$, $\textcolor{yellow}{\boldsymbol{\alpha}_{2}}$, $\textcolor{yellow}{\cdots}$, $\textcolor{yellow}{\boldsymbol{\alpha}_{m}}$ 线

选项

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

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

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

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


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