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\begin_body

\begin_layout Section
线性方程组与向量组的关系
\end_layout

\begin_layout Standard
\begin_inset Separator plain
\end_inset


\end_layout

\begin_layout Frame
\begin_inset Argument 4
status open

\begin_layout Plain Layout
简介
\end_layout

\end_inset


\end_layout

\begin_deeper
\begin_layout Standard
运用线性方程组讨论向量组的线性组合、线性相关性、线性表示以及等价。
\end_layout

\end_deeper
\begin_layout Subsection
线性方程组的向量表示形式
\end_layout

\begin_layout Frame
\begin_inset Argument 3
status open

\begin_layout Plain Layout
allowframebreaks
\end_layout

\end_inset


\begin_inset Argument 4
status open

\begin_layout Plain Layout
线性方程组的向量表示形式
\end_layout

\end_inset


\end_layout

\begin_deeper
\begin_layout Standard
线性方程组的向量表示形式如下
\begin_inset Formula 
\begin{align*}
x_{1}\alpha_{1}+x_{2}\alpha_{2}+\cdots+x_{n}\alpha_{n} & =b,\\
x_{1}\alpha_{1}+x_{2}\alpha_{2}+\cdots+x_{n}\alpha_{n} & =0,
\end{align*}

\end_inset

其中
\begin_inset Formula 
\[
\alpha_{j}=\begin{bmatrix}a_{1j}\\
a_{2j}\\
\vdots\\
a_{mj}
\end{bmatrix},\quad j=1,2,\cdots,n.
\]

\end_inset


\end_layout

\end_deeper
\begin_layout Standard
\begin_inset Separator plain
\end_inset


\end_layout

\begin_layout Frame
\begin_inset Argument 4
status open

\begin_layout Plain Layout
线性方程组与向量组的线性关系
\end_layout

\end_inset


\end_layout

\begin_deeper
\begin_layout Itemize
\begin_inset Formula $m$
\end_inset

 维向量 
\begin_inset Formula $b$
\end_inset

 是否可以由 
\begin_inset Formula $m$
\end_inset

 维向量组 
\begin_inset Formula $\alpha_{1},\alpha_{2},\cdots,\alpha_{n}$
\end_inset

 线性表示?
\end_layout

\begin_layout Itemize
非齐次线性方程组 
\begin_inset Formula $\begin{cases}
a_{11}x_{1}+a_{12}x_{2}+\cdots+a_{1n}x_{n}=b_{1},\\
a_{21}x_{1}+a_{22}x_{2}+\cdots+a_{2n}x_{n}=b_{2},\\
\vdots\\
a_{m1}x_{1}+a_{m2}x_{2}+\cdots+a_{mn}x_{n}=b_{m},
\end{cases}$
\end_inset

 是否有解?
\end_layout

\begin_layout Itemize
向量组 
\begin_inset Formula $\alpha_{1},\alpha_{2},\cdots,\alpha_{n}$
\end_inset

 是否线性相关?
\end_layout

\begin_layout Itemize
齐次线性方程组 
\begin_inset Formula $\begin{cases}
a_{11}x_{1}+a_{12}x_{2}+\cdots+a_{1n}x_{n}=0,\\
a_{21}x_{1}+a_{22}x_{2}+\cdots+a_{2n}x_{n}=0,\\
\vdots\\
a_{m1}x_{1}+a_{m2}x_{2}+\cdots+a_{mn}x_{n}=0,
\end{cases}$
\end_inset

 是否有非零解?
\end_layout

\end_deeper
\begin_layout Frame

\end_layout

\begin_layout Standard
\begin_inset Separator plain
\end_inset


\end_layout

\begin_layout Frame
\begin_inset Argument 4
status open

\begin_layout Plain Layout
非齐次线性方程组与向量组的线性表示
\end_layout

\end_inset


\end_layout

\begin_deeper
\begin_layout Standard
记 
\begin_inset Formula $A=\left(\alpha_{1},\alpha_{2},\cdots,\alpha_{n}\right)$
\end_inset

.
 
\end_layout

\begin_layout Corollary
(1).
 
\begin_inset Formula $m$
\end_inset

 维向量 
\begin_inset Formula $b$
\end_inset

 能由 
\begin_inset Formula $m$
\end_inset

 维向量组 
\begin_inset Formula $\alpha_{1},\alpha_{2},\cdots,\alpha_{n}$
\end_inset

 唯一线性表示的充要条件是: 非齐次线性方程组 
\begin_inset Formula $Ax=b$
\end_inset

 有唯一解, 即 
\begin_inset Formula $r(A)=r(A,b)=n$
\end_inset

.
\end_layout

\begin_layout Corollary
(2).
 
\begin_inset Formula $m$
\end_inset

 维向量 
\begin_inset Formula $b$
\end_inset

 能由 
\begin_inset Formula $m$
\end_inset

 维向量组 
\begin_inset Formula $\alpha_{1},\alpha_{2},\cdots,\alpha_{n}$
\end_inset

 线性表示, 且表示法不唯一的充要条件是: 非齐次线性方程组 
\begin_inset Formula $Ax=b$
\end_inset

 有无穷多解, 即 
\begin_inset Formula $r(A)=r(A,b)<n$
\end_inset

.
\end_layout

\begin_layout Corollary
(3).
 
\begin_inset Formula $m$
\end_inset

 维向量 
\begin_inset Formula $b$
\end_inset

 不能由 
\begin_inset Formula $m$
\end_inset

 维向量组 
\begin_inset Formula $\alpha_{1},\alpha_{2},\cdots,\alpha_{n}$
\end_inset

 线性表示的充要条件是: 非齐次线性方程组 
\begin_inset Formula $Ax=b$
\end_inset

 无解, 即 
\begin_inset Formula $r(A)<r(A,b)$
\end_inset

.
\end_layout

\begin_layout Standard
设向量组 
\begin_inset Formula $A:\alpha_{1},\alpha_{2},\cdots,\alpha_{m}$
\end_inset

, 向量组 
\begin_inset Formula $B:\beta_{1},\beta_{2},\cdots,\beta_{n}$
\end_inset

, 若向量组 
\begin_inset Formula $B$
\end_inset

 能由向量组 
\begin_inset Formula $A$
\end_inset

 线性表示, 则存在矩阵 
\begin_inset Formula $A_{m\times n}$
\end_inset

, 使 
\begin_inset Formula 
\[
\left(\beta_{1},\beta_{2},\cdots,\beta_{n}\right)=\left(\alpha_{1},\alpha_{2},\cdots,\alpha_{m}\right)A_{m\times n},
\]

\end_inset

即矩阵方程
\begin_inset Formula 
\[
AX=B\Longleftrightarrow\left(\alpha_{1},\alpha_{2},\cdots,\alpha_{m}\right)X=\left(\beta_{1},\beta_{2},\cdots,\beta_{n}\right)
\]

\end_inset

有解.
\end_layout

\end_deeper
\begin_layout Standard
\begin_inset Separator plain
\end_inset


\end_layout

\begin_layout Frame
\begin_inset Argument 4
status open

\begin_layout Plain Layout
向量组之间的线性表示
\end_layout

\end_inset


\end_layout

\begin_deeper
\begin_layout Standard
记
\begin_inset Formula 
\[
A=\left(\alpha_{1},\alpha_{2},\cdots,\alpha_{m}\right),\ B=\left(\beta_{1},\beta_{2},\cdots,\beta_{n}\right).
\]

\end_inset


\end_layout

\begin_layout Corollary
(1).
 向量组 
\begin_inset Formula $B:\beta_{1},\beta_{2},\cdots,\beta_{n}$
\end_inset

 能由向量组 
\begin_inset Formula $A:\alpha_{1},\alpha_{2},\cdots,\alpha_{m}$
\end_inset

 线性表示的充要条件是 
\begin_inset Formula $r(A)=r(A,B)$
\end_inset

.
\end_layout

\begin_layout Corollary
(2).
 向量组 
\begin_inset Formula $A:\alpha_{1},\alpha_{2},\cdots,\alpha_{m}$
\end_inset

 与向量组 
\begin_inset Formula $B:\beta_{1},\beta_{2},\cdots,\beta_{n}$
\end_inset

 等价的充要条件是 
\begin_inset Formula $r(A)=r(B)=r(A,B)$
\end_inset

.
\end_layout

\end_deeper
\begin_layout Standard
\begin_inset Separator plain
\end_inset


\end_layout

\begin_layout Frame
\begin_inset Argument 4
status open

\begin_layout Plain Layout
齐次线性方程组与向量组的线性表示
\end_layout

\end_inset


\end_layout

\begin_deeper
\begin_layout Corollary
(1).
 
\begin_inset Formula $m$
\end_inset

 维向量组 
\begin_inset Formula $\alpha_{1},\alpha_{2},\cdots,\alpha_{n}$
\end_inset

 线性线性相关的充要条件是: 齐次线性方程组 
\begin_inset Formula $Ax=0$
\end_inset

 有非零解, 即 
\begin_inset Formula $r(A)<n$
\end_inset

.
\end_layout

\begin_layout Corollary
(2).
 
\begin_inset Formula $m$
\end_inset

 维向量组 
\begin_inset Formula $\alpha_{1},\alpha_{2},\cdots,\alpha_{n}$
\end_inset

 线性无关的充要条件是: 齐次线性方程组 
\begin_inset Formula $Ax=0$
\end_inset

 只有唯一解, 即 
\begin_inset Formula $r(A)=n$
\end_inset

.
\end_layout

\end_deeper
\begin_layout Standard
\begin_inset Separator plain
\end_inset


\end_layout

\begin_layout Subsection
齐次方程组?
\end_layout

\begin_layout Frame
\begin_inset Argument 4
status open

\begin_layout Plain Layout
齐次方程组?
\end_layout

\end_inset


\end_layout

\begin_deeper
\begin_layout Example
设 
\begin_inset Formula $A=\begin{bmatrix}1 & 2 & -2\\
4 & t & 3\\
3 & -1 & 1
\end{bmatrix}$
\end_inset

, 
\begin_inset Formula $B_{3\times3}\neq O$
\end_inset

 且 
\begin_inset Formula $AB=O$
\end_inset

, 求 
\begin_inset Formula $t$
\end_inset

.
\end_layout

\begin_layout Solution*
因为 
\begin_inset Formula $B_{3\times3}\neq O$
\end_inset

 且 
\begin_inset Formula $AB=O$
\end_inset

, 所以 
\begin_inset Formula $Ax=0$
\end_inset

 有非零解, 由克莱姆法则知, 
\begin_inset Formula $|A|=0$
\end_inset

, 所以 
\begin_inset Formula $t=-3$
\end_inset

.
\end_layout

\end_deeper
\begin_layout Standard
\begin_inset Separator plain
\end_inset


\end_layout

\begin_layout Frame
\begin_inset Argument 3
status open

\begin_layout Plain Layout
allowframebreaks
\end_layout

\end_inset


\begin_inset Argument 4
status open

\begin_layout Plain Layout
线性方程组有解的判别定理
\end_layout

\end_inset


\end_layout

\begin_deeper
\begin_layout Example
设 
\begin_inset Formula $\beta=\begin{bmatrix}b_{1} & b_{2} & \cdots & b_{n}\end{bmatrix}^{T}$
\end_inset

 是齐次线性方程组 
\begin_inset Formula $Ax=0$
\end_inset

 的一个非零解, 其中 
\begin_inset Formula 
\[
A=\begin{bmatrix}a_{11} & a_{12} & \cdots & a_{1n}\\
a_{21} & a_{22} & \cdots & a_{2n}\\
\vdots & \vdots & \ddots & \vdots\\
a_{m1} & a_{m2} & \cdots & a_{mn}
\end{bmatrix},\quad m<n.
\]

\end_inset

令 
\begin_inset Formula $\alpha_{i}=\begin{bmatrix}a_{i1} & a_{i2} & \cdots & a_{in}\end{bmatrix}^{T}$
\end_inset

, (
\begin_inset Formula $i=1,2,\cdots,m$
\end_inset

).
 若 
\begin_inset Formula $\alpha_{1},\alpha_{2},\cdots,\alpha_{m}$
\end_inset

 线性无关, 试判断 
\begin_inset Formula $\alpha_{1},\alpha_{2},\cdots,\alpha_{m},\beta$
\end_inset

 的线性相关性.
\end_layout

\begin_layout Proof
根据线性相关/线性无关性的定义, 设常数 
\begin_inset Formula $k_{1},k_{2},\cdots,k_{m},k$
\end_inset

 使得
\begin_inset Formula 
\begin{equation}
k_{1}\alpha_{1}+k_{2}\alpha_{2}+\cdots+k_{m}\alpha_{m}+k\beta=0,\label{eq:2.2-1}
\end{equation}

\end_inset

下面验证这些 
\begin_inset Formula $k_{1},k_{2},\cdots,k_{m},k$
\end_inset

 是否只有零解 (对应向量组的线性无关性), 还是存在非零解 
\begin_inset Formula $k_{1},k_{2},\cdots,k_{m},k$
\end_inset

 (对应向量组的线性相关性).
\end_layout

\begin_layout Standard
\begin_inset Separator plain
\end_inset


\end_layout

\begin_layout Proof
由已知, 矩阵
\begin_inset Formula 
\[
A=\begin{bmatrix}a_{11} & a_{12} & \cdots & a_{1n}\\
a_{21} & a_{22} & \cdots & a_{2n}\\
\vdots & \vdots & \ddots & \vdots\\
a_{m1} & a_{m2} & \cdots & a_{mn}
\end{bmatrix}=\begin{bmatrix}\alpha_{1}^{T}\\
\alpha_{2}^{T}\\
\vdots\\
\alpha_{m}^{T}
\end{bmatrix},
\]

\end_inset

由于 
\begin_inset Formula $\beta$
\end_inset

 满足齐次线性方程组 
\begin_inset Formula $Ax=0$
\end_inset

, 因此
\begin_inset Formula 
\[
0=A\beta=\begin{bmatrix}\alpha_{1}^{T}\\
\alpha_{2}^{T}\\
\vdots\\
\alpha_{m}^{T}
\end{bmatrix}\beta=\begin{bmatrix}\alpha_{1}^{T}\beta\\
\alpha_{2}^{T}\beta\\
\vdots\\
\alpha_{m}^{T}\beta
\end{bmatrix}.
\]

\end_inset

于是对于任意的 
\begin_inset Formula $i=1,2,\cdots,m$
\end_inset

, 有 
\begin_inset Formula $\alpha_{i}^{T}\beta=0$
\end_inset

.
 
\end_layout

\begin_layout Standard
\begin_inset Separator plain
\end_inset


\end_layout

\begin_layout Proof
由方程 (
\begin_inset CommandInset ref
LatexCommand ref
reference "eq:2.2-1"
plural "false"
caps "false"
noprefix "false"

\end_inset

) 得
\begin_inset Formula 
\[
k_{1}\alpha_{1}^{T}+k_{2}\alpha_{2}^{T}+\cdots+k_{m}\alpha_{m}^{T}+k\beta^{T}=0,
\]

\end_inset

对上式左右两边同时右乘 
\begin_inset Formula $\beta$
\end_inset

, 得到 
\begin_inset Formula $k\beta^{T}\beta=0$
\end_inset

, 由于 
\begin_inset Formula $\beta\ne0$
\end_inset

, 所以必有 
\begin_inset Formula $k=0$
\end_inset

.
 于是上式变成
\begin_inset Formula 
\[
k_{1}\alpha_{1}^{T}+k_{2}\alpha_{2}^{T}+\cdots+k_{m}\alpha_{m}^{T}=0,
\]

\end_inset

最后由 
\begin_inset Formula $\alpha_{1},\alpha_{2},\cdots,\alpha_{m}$
\end_inset

 线性无关, 知 
\begin_inset Formula $k_{1}=k_{2}=\cdots=k_{m}=0$
\end_inset

.
 
\end_layout

\begin_layout Proof
所以向量组 
\begin_inset Formula $\alpha_{1},\alpha_{2},\cdots,\alpha_{m},\beta$
\end_inset

 线性无关.
\end_layout

\end_deeper
\begin_layout Frame

\end_layout

\end_body
\end_document