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\begin_body
\begin_layout Section
向量组的线性组合
\end_layout
\begin_layout Subsection
\begin_inset Formula $n$
\end_inset
维向量
\end_layout
\begin_layout Frame
\begin_inset Argument 4
status open
\begin_layout Plain Layout
\begin_inset Formula $n$
\end_inset
维向量及其线性运算
\end_layout
\end_inset
\end_layout
\begin_deeper
\begin_layout Definition
\begin_inset Formula $n$
\end_inset
个有次序的数
\begin_inset Formula $\left(a_{1},a_{2},\cdots,a_{n}\right)$
\end_inset
所组成的数组称为
\begin_inset Formula $n$
\end_inset
维向量, 这
\begin_inset Formula $n$
\end_inset
个数称为该
\series bold
向量的
\begin_inset Formula $n$
\end_inset
个分量
\series default
, 第
\begin_inset Formula $i$
\end_inset
个数
\begin_inset Formula $a_{i}$
\end_inset
称为
\series bold
第
\begin_inset Formula $i$
\end_inset
个分量
\series default
.
\end_layout
\begin_layout Itemize
在解析几何中, “既有大小又有方向的量” 称为
\series bold
向量
\series default
, 并把可随意平行移动的有向线段作为向量的几何形象.
\end_layout
\begin_layout Itemize
引入坐标系后, 又定义了向量的坐标表示式 (三个有序实数), 此即上面定义的
\begin_inset Formula $3$
\end_inset
维向量.
\end_layout
\begin_layout Itemize
因此, 当
\begin_inset Formula $n\leq3$
\end_inset
时,
\begin_inset Formula $n$
\end_inset
维向量可以把有向线段作为其几何形象.
当
\begin_inset Formula $n>3$
\end_inset
时,
\begin_inset Formula $n$
\end_inset
维向量没有直观的几何形象.
\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
列向量组与行向量组
\end_layout
\end_inset
\end_layout
\begin_deeper
\begin_layout Standard
若干个同维数的列向量 (或行向量) 所组成的集合称为
\series bold
向量组
\series default
.
例如, 一个
\begin_inset Formula $m\times n$
\end_inset
矩阵
\begin_inset Formula
\[
A=\begin{bmatrix}a_{11} & a_{12} & \cdots & a_{1n}\\
a_{21} & a_{22} & \cdots & a_{2n}\\
\vdots & \vdots & \ddots & \vdots\\
a_{m1} & a_{m2} & \cdots & a_{mn}
\end{bmatrix},
\]
\end_inset
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
vspace{-5mm}
\end_layout
\end_inset
\end_layout
\begin_layout Standard
每一列
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
vspace{-5mm}
\end_layout
\end_inset
\end_layout
\begin_layout Standard
\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
\begin_layout Standard
组成的向量组
\begin_inset Formula $\alpha_{1},\alpha_{2},\cdots,\alpha_{n}$
\end_inset
称为
\series bold
矩阵
\begin_inset Formula $A$
\end_inset
的列向量组
\series default
.
\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
列向量组与行向量组
\end_layout
\end_inset
\end_layout
\begin_deeper
\begin_layout Standard
而由矩阵
\begin_inset Formula $A$
\end_inset
的每一行
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
vspace{-5mm}
\end_layout
\end_inset
\end_layout
\begin_layout Standard
\begin_inset Formula
\[
\beta_{i}=\left(a_{i1},a_{i2},\cdots,a_{in}\right),\quad(i=1,2,\cdots,m)
\]
\end_inset
\end_layout
\begin_layout Standard
组成的向量组
\begin_inset Formula $\beta_{1},\beta_{2},\cdots,\beta_{m}$
\end_inset
称为
\series bold
矩阵
\begin_inset Formula $A$
\end_inset
的行向量组
\series default
.
根据上述讨论, 矩阵
\begin_inset Formula $A$
\end_inset
记为
\end_layout
\begin_layout Standard
\begin_inset Formula
\[
A=\left(\alpha_{1},\alpha_{2},\cdots,\alpha_{n}\right)\text{ 或 }A=\begin{bmatrix}\beta_{1}\\
\beta_{2}\\
\vdots\\
\beta_{n}
\end{bmatrix}.
\]
\end_inset
\end_layout
\begin_layout Standard
这样, 矩阵
\begin_inset Formula $A$
\end_inset
就与其
\series bold
列向量组
\series default
或
\series bold
行向量组
\series default
之间建立了一一对应关系.
\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
\begin_inset Formula $n$
\end_inset
维向量的线性运算
\end_layout
\end_inset
\end_layout
\begin_deeper
\begin_layout Definition
两个
\begin_inset Formula $n$
\end_inset
维向量
\begin_inset Formula $\alpha=\left(a_{1},a_{2},\cdots,a_{n}\right)$
\end_inset
与
\begin_inset Formula $\beta=\left(b_{1},b_{2},\cdots,b_{n}\right)$
\end_inset
的各对应分量之和组成的向量, 称为向量
\begin_inset Formula $\alpha$
\end_inset
与
\begin_inset Formula $\beta$
\end_inset
的
\series bold
和
\series default
, 记为
\begin_inset Formula $\alpha+\beta$
\end_inset
, 即
\begin_inset Formula
\[
\alpha+\beta=\left(a_{1}+b_{1},a_{2}+b_{2},\cdots,a_{n}+b_{n}\right).
\]
\end_inset
由加法和
\series bold
负向量的定义
\series default
, 可定义向量的减法:
\begin_inset Formula
\[
\begin{aligned}\alpha-\beta= & \alpha+(-\beta)\\
& =\left(a_{1}-b_{1},a_{2}-b_{2},\cdots,a_{n}-b_{n}\right).
\end{aligned}
\]
\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
\begin_inset Formula $n$
\end_inset
维向量的线性运算
\end_layout
\end_inset
\end_layout
\begin_deeper
\begin_layout Definition
\begin_inset Formula $n$
\end_inset
维向量
\begin_inset Formula $\alpha=\left(a_{1},a_{2},\cdots,a_{n}\right)$
\end_inset
的各个分量都乘以实数
\begin_inset Formula $k$
\end_inset
所组成的向量, 称为数
\begin_inset Formula $k$
\end_inset
与向量
\begin_inset Formula $\alpha$
\end_inset
的乘积 (又简称为
\series bold
数乘
\series default
), 记为
\begin_inset Formula $k\alpha$
\end_inset
, 即
\begin_inset Formula
\[
k\alpha=\left(ka_{1},ka_{2},\cdots,ka_{n}\right).
\]
\end_inset
\end_layout
\begin_layout Standard
向量的加法和数乘运算统称为
\series bold
向量的线性运算
\series default
.
\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
\begin_inset Formula $n$
\end_inset
维向量的运算规律
\end_layout
\end_inset
\end_layout
\begin_deeper
\begin_layout Remark*
向量的线性运算与行 (列) 矩阵的运算规律相同, 从而也满足下列运算规律:
\end_layout
\begin_layout Remark*
(1)
\begin_inset Formula $\alpha+\beta=\beta+\alpha$
\end_inset
;
\end_layout
\begin_layout Remark*
(2)
\begin_inset Formula $(\alpha+\beta)+\gamma=\alpha+(\beta+\gamma)$
\end_inset
;
\end_layout
\begin_layout Remark*
(3)
\begin_inset Formula $\alpha+0=\alpha$
\end_inset
;
\end_layout
\begin_layout Remark*
(4)
\begin_inset Formula $\alpha+(-\alpha)=0$
\end_inset
;
\end_layout
\begin_layout Remark*
(5)
\begin_inset Formula $1\alpha=\alpha$
\end_inset
;
\end_layout
\begin_layout Remark*
(6)
\begin_inset Formula $k(l\alpha)=(kl)\alpha$
\end_inset
;
\end_layout
\begin_layout Remark*
(7)
\begin_inset Formula $k(\alpha+\beta)=k\alpha+k\beta$
\end_inset
;
\end_layout
\begin_layout Remark*
(8)
\begin_inset Formula $(k+l)\alpha=k\alpha+l\alpha$
\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
求向量方程
\end_layout
\end_inset
\end_layout
\begin_deeper
\begin_layout Example
设
\begin_inset Formula $\alpha_{1}=\begin{bmatrix}2\\
-4\\
1\\
-1
\end{bmatrix}$
\end_inset
,
\begin_inset Formula $\alpha_{2}=\begin{bmatrix}-3\\
-1\\
2\\
-\frac{5}{2}
\end{bmatrix}$
\end_inset
, 如果向量满足
\begin_inset Formula $3\alpha_{1}-2\left(\beta+\alpha_{2}\right)=0$
\end_inset
, 求
\begin_inset Formula $\beta$
\end_inset
.
\end_layout
\begin_deeper
\begin_layout Pause
\end_layout
\end_deeper
\begin_layout Solution*
由题设条件, 有
\begin_inset Formula $3\alpha_{1}-2\beta-2\alpha_{2}=0$
\end_inset
, 所以有
\begin_inset Formula
\begin{align*}
\beta & =-\frac{1}{2}\left(2\alpha_{2}-3\alpha_{1}\right)=-\alpha_{2}+\frac{3}{2}\alpha_{1}\\
& =-\begin{bmatrix}-3\\
-1\\
2\\
-\frac{5}{2}
\end{bmatrix}+\frac{3}{2}\begin{bmatrix}2\\
-4\\
1\\
-1
\end{bmatrix}=\begin{bmatrix}6\\
-5\\
-\frac{1}{2}\\
1
\end{bmatrix}.
\end{align*}
\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
求向量方程
\end_layout
\end_inset
\end_layout
\begin_deeper
\begin_layout Example
设
\begin_inset Formula $\alpha=(2,0,-1,3)^{T}$
\end_inset
,
\begin_inset Formula $\beta=(1,7,4,-2)^{T}$
\end_inset
,
\begin_inset Formula $\gamma=(0,1,0,1)^{T}$
\end_inset
.
\end_layout
\begin_layout Example
(1) 求
\begin_inset Formula $2\alpha+\beta-3\gamma$
\end_inset
;
\end_layout
\begin_layout Example
(2) 若有
\begin_inset Formula $x$
\end_inset
, 满足
\begin_inset Formula $3\alpha-\beta+5\gamma+2x=0$
\end_inset
, 求
\begin_inset Formula $x$
\end_inset
.
\end_layout
\begin_deeper
\begin_layout Pause
\end_layout
\end_deeper
\begin_layout Solution*
(1)
\begin_inset Formula $2\alpha+\beta-3\gamma=2(2,0,-1,3)^{T}+(1,7,4,-2)^{T}-3(0,1,0,1)^{T}=(5,4,2,1)^{T}$
\end_inset
.
\end_layout
\begin_layout Solution*
(2) 由
\begin_inset Formula $3\alpha-\beta+5\gamma+2x=0$
\end_inset
, 得
\begin_inset Formula
\begin{align*}
x & =\frac{1}{2}(-3\alpha+\beta-5\gamma)\\
& =\frac{1}{2}\left[-3(2,0,-1,3)^{T}+(1,7,4,-2)^{T}-5(0,1,0,1)^{T}\right]=(-5/2,1,7/2,-8)^{T}.
\end{align*}
\end_inset
\end_layout
\end_deeper
\begin_layout Subsection
向量组的线性组合
\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组合
\end_layout
\end_inset
\end_layout
\begin_deeper
\begin_layout Standard
考察非齐次线性方程组
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
vspace{-4mm}
\end_layout
\end_inset
\begin_inset Formula
\begin{equation}
\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}\\
\cdots\cdots\cdots\cdots\cdots\cdots\\
a_{m1}x_{1}+a_{m2}x_{2}+\cdots+a_{mn}x_{n}=b_{m}
\end{cases}\label{eq:3.2-1}
\end{equation}
\end_inset
令
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
vspace{-4mm}
\end_layout
\end_inset
\begin_inset Formula
\[
\alpha_{j}=\begin{bmatrix}a_{1j}\\
a_{2j}\\
\vdots\\
a_{mj}
\end{bmatrix},\quad(j=1,2,\cdots,n),\quad\beta=\begin{bmatrix}b_{1}\\
b_{2}\\
\vdots\\
b_{m}
\end{bmatrix},
\]
\end_inset
则线性方程组 (
\begin_inset CommandInset ref
LatexCommand ref
reference "eq:3.2-1"
plural "false"
caps "false"
noprefix "false"
\end_inset
) 可表为如下向量形式:
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
vspace{-4mm}
\end_layout
\end_inset
\begin_inset Formula
\[
\alpha_{1}x_{1}+\alpha_{2}x_{2}+\cdots+\alpha_{n}x_{n}=\beta.
\]
\end_inset
于是, 线性方程组 (
\begin_inset CommandInset ref
LatexCommand ref
reference "eq:3.2-1"
plural "false"
caps "false"
noprefix "false"
\end_inset
) 是否有解, 就相当于是否存在一组数
\begin_inset Formula $k_{1},k_{2},\cdots,k_{n}$
\end_inset
使得下列线性关系式成立:
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
vspace{-4mm}
\end_layout
\end_inset
\begin_inset Formula
\[
\beta=k_{1}\alpha_{1}+k_{2}\alpha_{2}+\cdots+k_{n}\alpha_{n}.
\]
\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
向量组的线性组合与线性表示
\end_layout
\end_inset
\end_layout
\begin_deeper
\begin_layout Definition
给定向量组
\begin_inset Formula $A:\alpha_{1},\alpha_{2},\cdots,\alpha_{s}$
\end_inset
, 对于任何一组实数
\begin_inset Formula $k_{1},k_{2},\cdots,k_{s}$
\end_inset
, 表达式
\begin_inset Formula
\[
k_{1}\alpha_{1}+k_{2}\alpha_{2}+\cdots+k_{s}\alpha_{s}
\]
\end_inset
称为
\series bold
向量组
\begin_inset Formula $A$
\end_inset
的一个线性组合
\series default
,
\begin_inset Formula $k_{1},k_{2},\cdots,k_{s}$
\end_inset
称为这个
\series bold
线性组合的系数
\series default
.
\end_layout
\begin_layout Itemize
实数
\begin_inset Formula $k_{1},k_{2},\cdots,k_{s}$
\end_inset
: 线性组合的系数;
\end_layout
\begin_layout Itemize
\begin_inset Formula $k_{1}\alpha_{1}+k_{2}\alpha_{2}+\cdots+k_{s}\alpha_{s}$
\end_inset
: 向量组
\begin_inset Formula $A$
\end_inset
的一个线性表示.
\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
向量组的线性组合与线性表示
\end_layout
\end_inset
\end_layout
\begin_deeper
\begin_layout Definition
给定向量组
\begin_inset Formula $A:\alpha_{1},\alpha_{2},\cdots,\alpha_{s}$
\end_inset
和向量
\begin_inset Formula $\beta$
\end_inset
, 若存在一组数
\begin_inset Formula $k_{1},k_{2},\cdots,k_{s}$
\end_inset
, 使
\begin_inset Formula
\[
\beta=k_{1}\alpha_{1}+k_{2}\alpha_{2}+\cdots+k_{s}\alpha_{s},
\]
\end_inset
则称
\series bold
向量
\begin_inset Formula $\beta$
\end_inset
是向量组
\begin_inset Formula $A$
\end_inset
的线性组合
\series default
, 又称
\series bold
向量
\begin_inset Formula $\beta$
\end_inset
能由向量组
\begin_inset Formula $A$
\end_inset
线性表示
\series default
(或
\series bold
线性表出
\series default
).
\end_layout
\begin_layout Itemize
\begin_inset Formula $\beta=k_{1}\alpha_{1}+k_{2}\alpha_{2}+\cdots+k_{s}\alpha_{s}$
\end_inset
: 向量组
\begin_inset Formula $A$
\end_inset
的线性组合 (或线性表示)
\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
向量组的线性组合与线性方程组的关系
\end_layout
\end_inset
\end_layout
\begin_deeper
\begin_layout Remark*
(1)
\begin_inset Formula $\beta$
\end_inset
能由向量组
\begin_inset Formula $\alpha_{1},\alpha_{2},\cdots,\alpha_{s}$
\end_inset
唯一线性表示的充分必要条件是线性方程组
\begin_inset Formula
\[
\alpha_{1}x_{1}+\alpha_{2}x_{2}+\cdots+\alpha_{s}x_{s}=\beta
\]
\end_inset
有唯一解;
\end_layout
\begin_layout Remark*
(2)
\begin_inset Formula $\beta$
\end_inset
能由向量组
\begin_inset Formula $\alpha_{1},\alpha_{2},\cdots,\alpha_{s}$
\end_inset
线性表示且表示不唯一的充分必要条件是线性方程组
\begin_inset Formula
\[
\alpha_{1}x_{1}+\alpha_{2}x_{2}+\cdots+\alpha_{s}x_{s}=\beta
\]
\end_inset
有无穷多个解;
\end_layout
\begin_layout Remark*
(3)
\begin_inset Formula $\beta$
\end_inset
不能由向量组
\begin_inset Formula $\alpha_{1},\alpha_{2},\cdots,\alpha_{s}$
\end_inset
线性表示的充分必要条件是线性方程组
\begin_inset Formula
\[
\alpha_{1}x_{1}+\alpha_{2}x_{2}+\cdots+\alpha_{s}x_{s}=\beta
\]
\end_inset
无解;
\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
向量组的线性组合
\end_layout
\end_inset
\end_layout
\begin_deeper
\begin_layout Example
设
\begin_inset Formula $\alpha_{1}=(1,0,2,-1)$
\end_inset
,
\begin_inset Formula $\alpha_{2}=(3,0,4,1)$
\end_inset
,
\begin_inset Formula $\beta=(-1,0,0,-3)$
\end_inset
.
由于
\begin_inset Formula $\beta=2\alpha_{1}-\alpha_{2}$
\end_inset
, 因此
\begin_inset Formula $\beta$
\end_inset
是
\begin_inset Formula $\alpha_{1},\alpha_{2}$
\end_inset
的线性组合.
\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
向量组的线性组合
\end_layout
\end_inset
\end_layout
\begin_deeper
\begin_layout Example
证明: 向量
\begin_inset Formula $\beta=(-1,1,5)$
\end_inset
是向量
\begin_inset Formula $\alpha_{1}=(1,2,3)$
\end_inset
,
\begin_inset Formula $\alpha_{2}=(0,1,4)$
\end_inset
,
\begin_inset Formula $\alpha_{3}=(2,3,6)$
\end_inset
的线性组合并具体将
\begin_inset Formula $\beta$
\end_inset
用
\begin_inset Formula $\alpha_{1},\alpha_{2},\alpha_{3}$
\end_inset
表示出来.
\end_layout
\begin_layout Proof
先假定
\begin_inset Formula $\beta=\lambda_{1}\alpha_{1}+\lambda_{2}\alpha_{2}+\lambda_{3}\alpha_{3}$
\end_inset
, 其中
\begin_inset Formula $\lambda_{1},\lambda_{2},\lambda_{3}$
\end_inset
为待定常数, 则
\begin_inset Formula
\[
\begin{aligned}(-1,1,5) & =\lambda_{1}(1,2,3)+\lambda_{2}(0,1,4)+\lambda_{3}(2,3,6)\\
& =\left(\lambda_{1},2\lambda_{1},3\lambda_{1}\right)+\left(0,\lambda_{2},4\lambda_{2}\right)+\left(2\lambda_{3},3\lambda_{3},6\lambda_{3}\right)
\end{aligned}
\]
\end_inset
由于两个向量相等的充要条件是它们的分量分别对应相等, 因此可得方程组:
\begin_inset Formula
\[
\begin{cases}
\lambda_{1}+2\lambda_{3}=-1\\
2\lambda_{1}+\lambda_{2}+3\lambda_{3}=1\\
3\lambda_{1}+4\lambda_{2}+6\lambda_{3}=5
\end{cases}\Longrightarrow\begin{cases}
\lambda_{1}=1\\
\lambda_{2}=2\\
\lambda_{3}=-1
\end{cases}
\]
\end_inset
于是
\begin_inset Formula $\beta$
\end_inset
可以表示为
\begin_inset Formula $\alpha_{1},\alpha_{2},\alpha_{3}$
\end_inset
的线性组合, 它的表示式为
\begin_inset Formula $\beta=\alpha_{1}+2\alpha_{2}-\alpha_{3}$
\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
向量组的线性组合
\end_layout
\end_inset
\end_layout
\begin_deeper
\begin_layout Example
证明: 向量
\begin_inset Formula $(4,5,5)$
\end_inset
可以用多种方式表示成向量
\begin_inset Formula $(1,2,3)$
\end_inset
,
\begin_inset Formula $(-1,1,4)$
\end_inset
及
\begin_inset Formula $(3,3,2)$
\end_inset
的线性组合.
\end_layout
\begin_deeper
\begin_layout Pause
\end_layout
\end_deeper
\begin_layout Proof
假定
\begin_inset Formula $\lambda_{1},\lambda_{2},\lambda_{3}$
\end_inset
是数, 它们使
\begin_inset Formula
\[
\begin{aligned}(4,5,5) & =\lambda_{1}(1,2,3)+\lambda_{2}(-1,1,4)+\lambda_{3}(3,3,2)\\
& =\left(\lambda_{1},2\lambda_{1},3\lambda_{1}\right)+\left(-\lambda_{2},\lambda_{2},4\lambda_{2}\right)+\left(3\lambda_{3},3\lambda_{3},2\lambda_{3}\right)\\
& =\left(\lambda_{1}-\lambda_{2}+3\lambda_{3},2\lambda_{1}+\lambda_{2}+3\lambda_{3},3\lambda_{1}+4\lambda_{2}+2\lambda_{3}\right),
\end{aligned}
\]
\end_inset
这样便可得到一个线性方程组:
\begin_inset Formula
\begin{equation}
\begin{cases}
\lambda_{1}-\lambda_{2}+3\lambda_{3}=4\\
2\lambda_{1}+\lambda_{2}+3\lambda_{3}=5\\
3\lambda_{1}+4\lambda_{2}+2\lambda_{3}=5
\end{cases}\label{eq:3.2-2}
\end{equation}
\end_inset
这个方程组的解不是唯一的, 例如以下二组数都是方程组 (
\begin_inset CommandInset ref
LatexCommand ref
reference "eq:3.2-2"
plural "false"
caps "false"
noprefix "false"
\end_inset
) 的解:
\begin_inset Formula
\[
\lambda_{1}=1,\ \lambda_{2}=0,\ \lambda_{3}=1;\ \lambda_{1}=3,\ \lambda_{2}=-1,\ \lambda_{3}=0.
\]
\end_inset
因此
\begin_inset Formula $(4,5,5)=(1,2,3)+(3,3,2)$
\end_inset
;
\begin_inset Formula $(4,5,5)=3(1,2,3)-(-1,1,4)$
\end_inset
.
即向量
\begin_inset Formula $(4,5,5)$
\end_inset
可以用不止一种方式表示成另外
\begin_inset Formula $3$
\end_inset
个向量的线性组合.
\end_layout
\begin_layout Remark*
本例表明, 判断一个向量是否可用多种形式由其它向量组线性表出的问题, 也可以归结为某一个线性方程组解的个数问题.
\end_layout
\begin_layout Remark*
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
vspace{3mm}
\end_layout
\end_inset
\end_layout
\begin_layout Remark*
解唯一, 表示方式也唯一.
解越多, 表示方式也越多.
这说明线性方程组的解同向量线性关系之间的紧密联系.
\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
向量的线性组合
\end_layout
\end_inset
\end_layout
\begin_deeper
\begin_layout Example
任何一个
\begin_inset Formula $n$
\end_inset
维向量
\begin_inset Formula $\alpha=\left(a_{1},a_{2},\cdots,a_{n}\right)^{T}$
\end_inset
都是
\begin_inset Formula $n$
\end_inset
维单位向量组
\begin_inset Formula
\[
\varepsilon_{1}=\begin{bmatrix}1\\
0\\
0\\
\vdots\\
0
\end{bmatrix},\ \varepsilon_{2}=\begin{bmatrix}0\\
1\\
0\\
\vdots\\
0
\end{bmatrix},\ \cdots,\varepsilon_{n}=\begin{bmatrix}0\\
0\\
0\\
\vdots\\
1
\end{bmatrix}
\]
\end_inset
的线性组合.
因为
\begin_inset Formula $\alpha=a_{1}\varepsilon_{1}+a_{2}\varepsilon_{2}+\cdots+a_{n}\varepsilon_{n}$
\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
向量的线性组合
\end_layout
\end_inset
\end_layout
\begin_deeper
\begin_layout Example
零向量是任何一组向量的线性组合.
因为
\begin_inset Formula
\[
\boldsymbol{0}=0\cdot\alpha_{1}+0\cdot\alpha_{2}+\cdots+0\cdot\alpha_{s}.
\]
\end_inset
\end_layout
\begin_deeper
\begin_layout Pause
\end_layout
\end_deeper
\begin_layout Standard
\begin_inset Separator plain
\end_inset
\end_layout
\begin_layout Example
向量组
\begin_inset Formula $\alpha_{1},\alpha_{2},\cdots,\alpha_{s}$
\end_inset
中的任一向量
\begin_inset Formula $\alpha_{j}$
\end_inset
, (
\begin_inset Formula $1\leq j\leq s$
\end_inset
), 都是此向量组的线性组合.
因为
\begin_inset Formula
\[
\alpha_{j}=0\cdot\alpha_{1}+\cdots+1\cdot\alpha_{j}+\cdots+0\cdot\alpha_{s}.
\]
\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
向量组的线性组合
\end_layout
\end_inset
\end_layout
\begin_deeper
\begin_layout Example
判断向量
\begin_inset Formula $\beta=(4,3,-1,11)^{T}$
\end_inset
是否为向量组
\begin_inset Formula $\alpha_{1}=(1,2,-1,5)^{T}$
\end_inset
,
\begin_inset Formula $\alpha_{2}=(2,-1,1,1)^{T}$
\end_inset
的线性组合.
若是, 写出表示式.
\end_layout
\begin_layout Pause
\end_layout
\begin_layout Standard
\begin_inset Separator plain
\end_inset
\end_layout
\begin_layout Solution*
设
\begin_inset Formula $k_{1}\alpha_{1}+k_{2}\alpha_{2}=\beta$
\end_inset
, 对矩阵
\begin_inset Formula $\begin{bmatrix}\alpha_{1} & \alpha_{2} & \beta\end{bmatrix}$
\end_inset
做初等行变换:
\begin_inset Formula
\[
\begin{bmatrix}1 & 2 & 4\\
2 & -1 & 3\\
-1 & 1 & -1\\
5 & 1 & 11
\end{bmatrix}\longrightarrow\begin{bmatrix}1 & 2 & 4\\
0 & -5 & -5\\
0 & 3 & 3\\
0 & -9 & -9
\end{bmatrix}\longrightarrow\begin{bmatrix}1 & 2 & 4\\
0 & 1 & 1\\
0 & 0 & 0\\
0 & 0 & 0
\end{bmatrix}\longrightarrow{\color{red}\begin{bmatrix}1 & 0 & 2\\
0 & 1 & 1\\
0 & 0 & 0\\
0 & 0 & 0
\end{bmatrix}},
\]
\end_inset
故
\begin_inset Formula $\beta$
\end_inset
可由
\begin_inset Formula $\alpha_{1},\alpha_{2}$
\end_inset
线性表示, 且由上面的初等变换可取
\begin_inset Formula $k_{1}=2$
\end_inset
,
\begin_inset Formula $k_{2}=1$
\end_inset
使
\begin_inset Formula $\beta=2\alpha_{1}+\alpha_{2}$
\end_inset
.
\end_layout
\begin_layout Standard
由上面的阶梯型矩阵, 易见, 最后一列可以通过初等列变换由前面的列线性表示, 因此秩
\begin_inset Formula $r\begin{bmatrix}\alpha_{1} & \alpha_{2} & \beta\end{bmatrix}=r\begin{bmatrix}\alpha_{1} & \alpha_{2}\end{bmatrix}=2$
\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
上述算法的简短说明
\end_layout
\end_inset
\end_layout
\begin_deeper
\begin_layout Standard
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
begin{align*}
\end_layout
\begin_layout Plain Layout
\backslash
beta=k_{1}
\backslash
alpha_{1}+k_{2}
\backslash
alpha_{2} &
\backslash
Longleftrightarrow
\backslash
begin{bmatrix}4
\backslash
\backslash
3
\backslash
\backslash
-1
\backslash
\backslash
11
\backslash
end{bmatrix}=k_{1}
\backslash
begin{bmatrix}1
\backslash
\backslash
2
\backslash
\backslash
-1
\backslash
\backslash
5
\backslash
end{bmatrix}+k_{2}
\backslash
begin{bmatrix}2
\backslash
\backslash
-1
\backslash
\backslash
1
\backslash
\backslash
1
\backslash
end{bmatrix}
\backslash
\backslash
\end_layout
\begin_layout Plain Layout
&
\backslash
Longleftrightarrow
\backslash
begin{cases}k_{1}+2k_{2}=4,
\backslash
\backslash
2k_{1}-k_{2}=3,
\backslash
\backslash
-k_{1}+k_{2}=-1,
\backslash
\backslash
5k_{1}+k_{2}=11,
\backslash
end{cases}
\backslash
longleftrightarrow
\backslash
begin{bNiceMatrix}[first-row]k_{1} & k_{2} & 1
\backslash
\backslash
1 & 2 & 4
\backslash
\backslash
2 & -1 & 3
\backslash
\backslash
-1 & 1 & -1
\backslash
\backslash
5 & 1 & 11
\backslash
end{bNiceMatrix}
\backslash
\backslash
\end_layout
\begin_layout Plain Layout
&
\backslash
Longleftrightarrow
\backslash
begin{cases}k_{1}+2k_{2}=4,
\backslash
\backslash
-5k_{2}=-5,
\backslash
\backslash
3k_{2}=3,
\backslash
\backslash
-9k_{2}=-9,
\backslash
end{cases}
\backslash
longleftrightarrow
\backslash
begin{bNiceMatrix}[first-row]k_{1} & k_{2} & 1
\backslash
\backslash
1 & 2 & 4
\backslash
\backslash
0 & -5 & -5
\backslash
\backslash
0 & 3 & 3
\backslash
\backslash
0 & -9 & -9
\backslash
end{bNiceMatrix}.
\end_layout
\begin_layout Plain Layout
\backslash
end{align*}
\end_layout
\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
向量组的线性组合
\end_layout
\end_inset
\end_layout
\begin_deeper
\begin_layout Theorem
\begin_inset Argument 1
status open
\begin_layout Plain Layout
Rouché–Capelli 定理, Kronecker–Capelli 定理, Rouché–Fontené 定理, Rouché–Frobenius
定理, Frobenius 定理
\end_layout
\end_inset
设向量
\begin_inset Formula
\[
\beta=\begin{bmatrix}b_{1}\\
b_{2}\\
\vdots\\
b_{m}
\end{bmatrix},\quad\alpha_{j}=\begin{bmatrix}a_{1j}\\
a_{2j}\\
\vdots\\
a_{mj}
\end{bmatrix},\quad(j=1,2,\cdots,s),
\]
\end_inset
则向量
\begin_inset Formula $\beta$
\end_inset
能由向量组
\begin_inset Formula $\alpha_{1},\alpha_{2},\cdots,\alpha_{s}$
\end_inset
线性表示的充分必要条件是矩阵
\begin_inset Formula $A=\left(\alpha_{1},\alpha_{2},\cdots,\alpha_{s}\right)$
\end_inset
与矩阵
\begin_inset Formula $\widetilde{A}=\left(\alpha_{1},\alpha_{2},\cdots,\alpha_{s},\beta\right)$
\end_inset
的秩相等.
\end_layout
\end_deeper
\begin_layout Subsection
向量组间的线性表示
\end_layout
\begin_layout Frame
\begin_inset Argument 4
status open
\begin_layout Plain Layout
向量组间的线性表示
\end_layout
\end_inset
\end_layout
\begin_deeper
\begin_layout Definition
\begin_inset CommandInset label
LatexCommand label
name "def:3.1-16"
\end_inset
设有两向量组
\begin_inset Formula
\[
A:\alpha_{1},\alpha_{2},\cdots,\alpha_{s};\quad B:\beta_{1},\beta_{2},\cdots,\beta_{t},
\]
\end_inset
若向量组
\begin_inset Formula $B$
\end_inset
中的每一个向量都能由向量组
\begin_inset Formula $A$
\end_inset
线性表示, 则称
\series bold
向量组
\begin_inset Formula $B$
\end_inset
能由向量组
\begin_inset Formula $A$
\end_inset
线性表示
\series default
.
\end_layout
\begin_layout Definition
若向量组
\begin_inset Formula $A$
\end_inset
与向量组
\begin_inset Formula $B$
\end_inset
能相互线性表示, 则称这
\series bold
两个向量组等价
\series default
.
\end_layout
\begin_layout Standard
按定义, 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
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
vspace{-4mm}
\end_layout
\end_inset
\begin_inset Formula
\[
k_{1j},k_{2j},\cdots,k_{sj},\qquad(j=1,2,\cdots,t)
\]
\end_inset
使
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
vspace{-4mm}
\end_layout
\end_inset
\begin_inset Formula
\[
\beta_{j}=k_{1j}\alpha_{1}+k_{2j}\alpha_{2}+\cdots+k_{sj}\alpha_{s}=\left(\alpha_{1},\alpha_{2},\cdots,\alpha_{s}\right)\begin{bmatrix}k_{1j}\\
k_{2j}\\
\vdots\\
k_{sj}
\end{bmatrix},
\]
\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
向量组的线性表示
\end_layout
\end_inset
\end_layout
\begin_deeper
\begin_layout Overprint
\begin_inset Argument item:1
status open
\begin_layout Plain Layout
1
\end_layout
\end_inset
\begin_inset ERT
status open
\begin_layout Plain Layout
$$
\end_layout
\begin_layout Plain Layout
\backslash
colorbox{red!15}{$
\backslash
beta_{j}$}=k_{1j}
\backslash
alpha_{1}+k_{2j}
\backslash
alpha_{2}+
\backslash
cdots+k_{sj}
\backslash
alpha_{s}=
\end_layout
\begin_layout Plain Layout
\backslash
begin{bNiceMatrix}
\end_layout
\begin_layout Plain Layout
\backslash
alpha_{1} &
\backslash
alpha_{2} &
\backslash
cdots &
\backslash
alpha_{s}
\backslash
\backslash
\end_layout
\begin_layout Plain Layout
\backslash
CodeAfter
\backslash
tikz
\backslash
node [highlightd = (1-1) (1-4)] {} ;
\end_layout
\begin_layout Plain Layout
\backslash
end{bNiceMatrix}
\backslash
begin{bNiceMatrix}
\end_layout
\begin_layout Plain Layout
k_{1j}
\backslash
\backslash
\end_layout
\begin_layout Plain Layout
k_{2j}
\backslash
\backslash
\end_layout
\begin_layout Plain Layout
\backslash
vdots
\backslash
\backslash
\end_layout
\begin_layout Plain Layout
k_{sj}
\backslash
\backslash
\end_layout
\begin_layout Plain Layout
\backslash
CodeAfter
\backslash
tikz
\backslash
node [highlighta = (1-1) (4-1)] {} ;
\end_layout
\begin_layout Plain Layout
\backslash
end{bNiceMatrix},
\backslash
quad
\backslash
colorbox{red!15}{$j=1$}.
\end_layout
\begin_layout Plain Layout
$$
\end_layout
\end_inset
\end_layout
\begin_layout Overprint
\begin_inset Argument item:1
status open
\begin_layout Plain Layout
2
\end_layout
\end_inset
\begin_inset ERT
status open
\begin_layout Plain Layout
$$
\end_layout
\begin_layout Plain Layout
\backslash
colorbox{blue!15}{$
\backslash
beta_{j}$}=k_{1j}
\backslash
alpha_{1}+k_{2j}
\backslash
alpha_{2}+
\backslash
cdots+k_{sj}
\backslash
alpha_{s}=
\end_layout
\begin_layout Plain Layout
\backslash
begin{bNiceMatrix}
\end_layout
\begin_layout Plain Layout
\backslash
alpha_{1} &
\backslash
alpha_{2} &
\backslash
cdots &
\backslash
alpha_{s}
\backslash
\backslash
\end_layout
\begin_layout Plain Layout
\backslash
CodeAfter
\backslash
tikz
\backslash
node [highlightd = (1-1) (1-4)] {} ;
\end_layout
\begin_layout Plain Layout
\backslash
end{bNiceMatrix}
\backslash
begin{bNiceMatrix}
\end_layout
\begin_layout Plain Layout
k_{1j}
\backslash
\backslash
\end_layout
\begin_layout Plain Layout
k_{2j}
\backslash
\backslash
\end_layout
\begin_layout Plain Layout
\backslash
vdots
\backslash
\backslash
\end_layout
\begin_layout Plain Layout
k_{sj}
\backslash
\backslash
\end_layout
\begin_layout Plain Layout
\backslash
CodeAfter
\backslash
tikz
\backslash
node [highlightb = (1-1) (4-1)] {} ;
\end_layout
\begin_layout Plain Layout
\backslash
end{bNiceMatrix},
\backslash
quad
\backslash
colorbox{blue!15}{$j=2$}.
\end_layout
\begin_layout Plain Layout
$$
\end_layout
\end_inset
\end_layout
\begin_layout Overprint
\begin_inset Argument item:1
status open
\begin_layout Plain Layout
3
\end_layout
\end_inset
\begin_inset ERT
status open
\begin_layout Plain Layout
$$
\end_layout
\begin_layout Plain Layout
\backslash
colorbox{red!30}{$
\backslash
beta_{j}$}=k_{1j}
\backslash
alpha_{1}+k_{2j}
\backslash
alpha_{2}+
\backslash
cdots+k_{sj}
\backslash
alpha_{s}=
\end_layout
\begin_layout Plain Layout
\backslash
begin{bNiceMatrix}
\end_layout
\begin_layout Plain Layout
\backslash
alpha_{1} &
\backslash
alpha_{2} &
\backslash
cdots &
\backslash
alpha_{s}
\backslash
\backslash
\end_layout
\begin_layout Plain Layout
\backslash
CodeAfter
\backslash
tikz
\backslash
node [highlightd = (1-1) (1-4)] {} ;
\end_layout
\begin_layout Plain Layout
\backslash
end{bNiceMatrix}
\backslash
begin{bNiceMatrix}
\end_layout
\begin_layout Plain Layout
k_{1j}
\backslash
\backslash
\end_layout
\begin_layout Plain Layout
k_{2j}
\backslash
\backslash
\end_layout
\begin_layout Plain Layout
\backslash
vdots
\backslash
\backslash
\end_layout
\begin_layout Plain Layout
k_{sj}
\backslash
\backslash
\end_layout
\begin_layout Plain Layout
\backslash
CodeAfter
\backslash
tikz
\backslash
node [highlightc = (1-1) (4-1)] {} ;
\end_layout
\begin_layout Plain Layout
\backslash
end{bNiceMatrix},
\backslash
quad
\backslash
colorbox{red!30}{$j=t$}.
\end_layout
\begin_layout Plain Layout
$$
\end_layout
\end_inset
\end_layout
\begin_layout Standard
所以
\end_layout
\begin_layout Overprint
\begin_inset Argument item:1
status open
\begin_layout Plain Layout
1
\end_layout
\end_inset
\begin_inset ERT
status open
\begin_layout Plain Layout
$$
\end_layout
\begin_layout Plain Layout
\backslash
begin{bNiceMatrix}
\end_layout
\begin_layout Plain Layout
\backslash
beta_{1} &
\backslash
beta_{2} &
\backslash
cdots &
\backslash
beta_{t}
\backslash
\backslash
\end_layout
\begin_layout Plain Layout
\backslash
CodeAfter
\backslash
tikz
\backslash
node [highlighta = (1-1) (1-1)] {} ;
\end_layout
\begin_layout Plain Layout
\backslash
end{bNiceMatrix}=
\backslash
begin{bNiceMatrix}
\end_layout
\begin_layout Plain Layout
\backslash
alpha_{1} &
\backslash
alpha_{2} &
\backslash
cdots &
\backslash
alpha_{s}
\backslash
\backslash
\end_layout
\begin_layout Plain Layout
\backslash
CodeAfter
\backslash
tikz
\backslash
node [highlightd = (1-1) (1-4)] {} ;
\end_layout
\begin_layout Plain Layout
\backslash
end{bNiceMatrix}
\backslash
cdot
\backslash
begin{bNiceMatrix}
\end_layout
\begin_layout Plain Layout
k_{11}&k_{12}&
\backslash
cdots&k_{1t}
\backslash
\backslash
\end_layout
\begin_layout Plain Layout
k_{21}&k_{22}&
\backslash
cdots&k_{2t}
\backslash
\backslash
\end_layout
\begin_layout Plain Layout
\backslash
vdots&
\backslash
vdots&
\backslash
ddots&
\backslash
vdots
\backslash
\backslash
\end_layout
\begin_layout Plain Layout
k_{s1}&k_{s2}&
\backslash
cdots&k_{st}
\backslash
\backslash
\end_layout
\begin_layout Plain Layout
\backslash
CodeAfter
\backslash
tikz
\backslash
node [highlighta = (1-1) (4-1)] {} ;
\end_layout
\begin_layout Plain Layout
\backslash
end{bNiceMatrix},
\end_layout
\begin_layout Plain Layout
$$
\end_layout
\end_inset
\end_layout
\begin_layout Overprint
\begin_inset Argument item:1
status open
\begin_layout Plain Layout
2
\end_layout
\end_inset
\begin_inset ERT
status open
\begin_layout Plain Layout
$$
\end_layout
\begin_layout Plain Layout
\backslash
begin{bNiceMatrix}
\end_layout
\begin_layout Plain Layout
\backslash
beta_{1} &
\backslash
beta_{2} &
\backslash
cdots &
\backslash
beta_{t}
\backslash
\backslash
\end_layout
\begin_layout Plain Layout
\backslash
CodeAfter
\backslash
tikz
\backslash
node [highlightb = (1-2) (1-2)] {} ;
\end_layout
\begin_layout Plain Layout
\backslash
end{bNiceMatrix}=
\backslash
begin{bNiceMatrix}
\end_layout
\begin_layout Plain Layout
\backslash
alpha_{1} &
\backslash
alpha_{2} &
\backslash
cdots &
\backslash
alpha_{s}
\backslash
\backslash
\end_layout
\begin_layout Plain Layout
\backslash
CodeAfter
\backslash
tikz
\backslash
node [highlightd = (1-1) (1-4)] {} ;
\end_layout
\begin_layout Plain Layout
\backslash
end{bNiceMatrix}
\backslash
cdot
\backslash
begin{bNiceMatrix}
\end_layout
\begin_layout Plain Layout
k_{11}&k_{12}&
\backslash
cdots&k_{1t}
\backslash
\backslash
\end_layout
\begin_layout Plain Layout
k_{21}&k_{22}&
\backslash
cdots&k_{2t}
\backslash
\backslash
\end_layout
\begin_layout Plain Layout
\backslash
vdots&
\backslash
vdots&
\backslash
ddots&
\backslash
vdots
\backslash
\backslash
\end_layout
\begin_layout Plain Layout
k_{s1}&k_{s2}&
\backslash
cdots&k_{st}
\backslash
\backslash
\end_layout
\begin_layout Plain Layout
\backslash
CodeAfter
\backslash
tikz
\backslash
node [highlightb = (1-2) (4-2)] {} ;
\end_layout
\begin_layout Plain Layout
\backslash
end{bNiceMatrix},
\end_layout
\begin_layout Plain Layout
$$
\end_layout
\end_inset
\end_layout
\begin_layout Overprint
\begin_inset Argument item:1
status open
\begin_layout Plain Layout
3
\end_layout
\end_inset
\begin_inset ERT
status open
\begin_layout Plain Layout
$$
\end_layout
\begin_layout Plain Layout
\backslash
begin{bNiceMatrix}
\end_layout
\begin_layout Plain Layout
\backslash
beta_{1} &
\backslash
beta_{2} &
\backslash
cdots &
\backslash
beta_{t}
\backslash
\backslash
\end_layout
\begin_layout Plain Layout
\backslash
CodeAfter
\backslash
tikz
\backslash
node [highlightc = (1-4) (1-4)] {} ;
\end_layout
\begin_layout Plain Layout
\backslash
end{bNiceMatrix}=
\backslash
begin{bNiceMatrix}
\end_layout
\begin_layout Plain Layout
\backslash
alpha_{1} &
\backslash
alpha_{2} &
\backslash
cdots &
\backslash
alpha_{s}
\backslash
\backslash
\end_layout
\begin_layout Plain Layout
\backslash
CodeAfter
\backslash
tikz
\backslash
node [highlightd = (1-1) (1-4)] {} ;
\end_layout
\begin_layout Plain Layout
\backslash
end{bNiceMatrix}
\backslash
cdot
\backslash
begin{bNiceMatrix}
\end_layout
\begin_layout Plain Layout
k_{11}&k_{12}&
\backslash
cdots&k_{1t}
\backslash
\backslash
\end_layout
\begin_layout Plain Layout
k_{21}&k_{22}&
\backslash
cdots&k_{2t}
\backslash
\backslash
\end_layout
\begin_layout Plain Layout
\backslash
vdots&
\backslash
vdots&
\backslash
ddots&
\backslash
vdots
\backslash
\backslash
\end_layout
\begin_layout Plain Layout
k_{s1}&k_{s2}&
\backslash
cdots&k_{st}
\backslash
\backslash
\end_layout
\begin_layout Plain Layout
\backslash
CodeAfter
\backslash
tikz
\backslash
node [highlightc = (1-4) (4-4)] {} ;
\end_layout
\begin_layout Plain Layout
\backslash
end{bNiceMatrix},
\end_layout
\begin_layout Plain Layout
$$
\end_layout
\end_inset
\end_layout
\begin_layout Standard
其中矩阵
\begin_inset Formula $K_{s\times t}=\left(k_{ij}\right)_{s\times t}$
\end_inset
称为这一
\series bold
线性表示的系数矩阵
\series default
.
\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
向量组间的线性表示
\end_layout
\end_inset
\end_layout
\begin_deeper
\begin_layout Lemma
若
\begin_inset Formula $C_{s\times n}=A_{s\times t}B_{t\times n}$
\end_inset
, 则矩阵
\begin_inset Formula $C$
\end_inset
的列向量组能由矩阵
\begin_inset Formula $A$
\end_inset
的列向量组线性表示,
\begin_inset Formula $B$
\end_inset
为这一
\series bold
表示的系数矩阵
\series default
.
而矩阵
\begin_inset Formula $C$
\end_inset
的行向量组能由
\begin_inset Formula $B$
\end_inset
的行向量组线性表示,
\begin_inset Formula $A$
\end_inset
为这一表示的系数矩阵.
\end_layout
\begin_layout Theorem
若向量组
\begin_inset Formula $A$
\end_inset
可由向量组
\begin_inset Formula $B$
\end_inset
线性表示, 向量组
\begin_inset Formula $B$
\end_inset
可由向量组
\begin_inset Formula $C$
\end_inset
线性表示, 则向量组
\begin_inset Formula $A$
\end_inset
可由向量组
\begin_inset Formula $C$
\end_inset
线性表示.
\end_layout
\end_deeper
\begin_layout Standard
\begin_inset Separator plain
\end_inset
\end_layout
\begin_layout Subsection
作业
\end_layout
\begin_layout Frame
\begin_inset Argument 4
status open
\begin_layout Plain Layout
作业
\end_layout
\end_inset
\end_layout
\begin_deeper
\begin_layout Problem*
下列向量组中, 向量
\begin_inset Formula $\beta$
\end_inset
能否由其余向量线性表示? 若能, 写出线性表示式:
\begin_inset Formula
\[
\alpha_{1}=(3,-3,2)^{T},\ \alpha_{2}=(-2,1,2)^{T},\ \alpha_{3}=(1,2,-1)^{T},\ \beta=(4,5,6)^{T}.
\]
\end_inset
\end_layout
\begin_layout Standard
\begin_inset Separator plain
\end_inset
\end_layout
\begin_layout Problem*
判断下列行向量组是否线性相关.
\end_layout
\begin_layout Problem*
(1).
\begin_inset Formula $(1,2,3)$
\end_inset
,
\begin_inset Formula $(4,8,12)$
\end_inset
,
\begin_inset Formula $(3,0,1)$
\end_inset
,
\begin_inset Formula $(4,5,8)$
\end_inset
;
\end_layout
\begin_layout Problem*
(2).
\begin_inset Formula $(1,2,3,4,5,6)$
\end_inset
,
\begin_inset Formula $(1,0,1,0,1,0)$
\end_inset
,
\begin_inset Formula $(-1,1,1,-1,1,1)$
\end_inset
,
\begin_inset Formula $(-2,3,2,3,4,7)$
\end_inset
;
\end_layout
\begin_layout Problem*
(3).
\begin_inset Formula $(1,2,3,4)$
\end_inset
,
\begin_inset Formula $(1,0,1,0)$
\end_inset
,
\begin_inset Formula $(-1,1,1,-1)$
\end_inset
,
\begin_inset Formula $(-2,3,2,3)$
\end_inset
;
\end_layout
\begin_layout Problem*
(4).
\begin_inset Formula $(1,0,0,2,3,1)$
\end_inset
,
\begin_inset Formula $(0,1,0,4,6,2)$
\end_inset
,
\begin_inset Formula $(0,0,1,-2,-3,-1)$
\end_inset
;
\end_layout
\begin_layout Standard
\begin_inset Separator plain
\end_inset
\end_layout
\begin_layout Problem*
求满足下列等式的行向量
\begin_inset Formula $x$
\end_inset
:
\end_layout
\begin_layout Problem*
(1).
\begin_inset Formula $\alpha_{1}+2\alpha_{2}+3\alpha_{3}+4x=0$
\end_inset
, 其中
\begin_inset Formula $\alpha_{1}=(5,-8,-1,2)$
\end_inset
,
\begin_inset Formula $\alpha_{2}=(2,-1,4,-3)$
\end_inset
,
\begin_inset Formula $\alpha_{3}=(-3,2,-5,4)$
\end_inset
.
\end_layout
\begin_layout Problem*
(2).
\begin_inset Formula $3\left(\alpha_{1}-x\right)+2\left(\alpha_{2}+x\right)=5\left(\alpha_{3}+x\right)$
\end_inset
, 其中
\begin_inset Formula $\alpha_{1}=(2,5,1,3)$
\end_inset
,
\begin_inset Formula $\alpha_{2}=(10,1,5,10)$
\end_inset
,
\begin_inset Formula $\alpha_{3}=(4,1,-1,1)$
\end_inset
.
\end_layout
\end_deeper
\begin_layout Frame
\end_layout
\end_body
\end_document
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