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#LyX 2.3 created this file. For more info see http://www.lyx.org/
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\end_header
\begin_body
\begin_layout Section
向量组的秩
\end_layout
\begin_layout Subsection
极大线性无关向量组
\end_layout
\begin_layout Frame
\begin_inset Argument 4
status open
\begin_layout Plain Layout
极大线性无关向量组
\end_layout
\end_inset
\end_layout
\begin_deeper
\begin_layout Definition
设有向量组
\begin_inset Formula $A:\alpha_{1},\alpha_{2},\cdots,\alpha_{s}$
\end_inset
, 若在向量组
\begin_inset Formula $A$
\end_inset
中能选出
\begin_inset Formula $r$
\end_inset
个向量
\begin_inset Formula $\alpha_{1},\alpha_{2},\cdots,\alpha_{r}$
\end_inset
, 满足
\end_layout
\begin_layout Definition
(1) 向量组
\begin_inset Formula $A_{0}:\alpha_{1},\alpha_{2},\cdots,\alpha_{r}$
\end_inset
线性无关;
\end_layout
\begin_layout Definition
(2) 向量组
\begin_inset Formula $A$
\end_inset
中任意
\begin_inset Formula $r+1$
\end_inset
个向量 (若有的话) 都线性相关.
\end_layout
\begin_layout Definition
则称
\series bold
向量组
\begin_inset Formula $A_{0}$
\end_inset
是向量组
\begin_inset Formula $A$
\end_inset
的一个极大线性无关向量组
\series default
(简称为
\series bold
极大无关组
\series default
).
\end_layout
\begin_deeper
\begin_layout Pause
\end_layout
\end_deeper
\begin_layout Remark*
(1)
\series bold
\color orange
只含
\series default
有零向量的向量组没有极大无关组;
\end_layout
\begin_layout Remark*
(2) 向量组的极大无关组可能不止一个，但由上节推论
\begin_inset CommandInset ref
LatexCommand ref
reference "cor:3.2-6"
plural "false"
caps "false"
noprefix "false"
\end_inset
知, 其向量的个数是相同的.
\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
极大线性无关向量组
\end_layout
\end_inset
\end_layout
\begin_deeper
\begin_layout Theorem
\begin_inset CommandInset label
LatexCommand label
name "thm:3.4-1"
\end_inset
如果
\begin_inset Formula $\alpha_{j_{1}},\alpha_{j_{2}},\cdots,\alpha_{j_{r}}$
\end_inset
是
\begin_inset Formula $\alpha_{1},\alpha_{2},\cdots,\alpha_{s}$
\end_inset
的线性无关部分组, 它是极大无关组的充分必要条件是
\begin_inset Formula $\alpha_{1},\alpha_{2},\cdots,\alpha_{s}$
\end_inset
中的每一个向量都可由
\begin_inset Formula $\alpha_{j_{1}},\alpha_{j_{2}},\cdots,\alpha_{j_{r}}$
\end_inset
线性表示.
\end_layout
\begin_deeper
\begin_layout Pause
\end_layout
\end_deeper
\begin_layout Remark*
由定理
\begin_inset CommandInset ref
LatexCommand ref
reference "thm:3.4-1"
plural "false"
caps "false"
noprefix "false"
\end_inset
知, 向量组与其极大线性无关组可相互线性表示, 即向量组与其极大线性无关组等价 (向量组之间的等价见定义
\begin_inset CommandInset ref
LatexCommand ref
reference "def:3.1-16"
plural "false"
caps "false"
noprefix "false"
\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
极大线性无关向量组
\end_layout
\end_inset
\end_layout
\begin_deeper
\begin_layout Example
全体
\begin_inset Formula $n$
\end_inset
维向量
\begin_inset Formula $e_{1},e_{2},\cdots,e_{n}$
\end_inset
构成的向量组记作
\begin_inset Formula $\RR^{n}$
\end_inset
, 求
\begin_inset Formula $\RR^{n}$
\end_inset
的一个极大无关组及
\begin_inset Formula $\RR^{n}$
\end_inset
的秩.
\end_layout
\begin_layout Solution*
因为
\begin_inset Formula $n$
\end_inset
维单位坐标向量构成的向量组
\begin_inset Formula $E:e_{1},e_{2},\cdots,e_{n}$
\end_inset
是线性无关的, 又知,
\begin_inset Formula $\RR^{n}$
\end_inset
中的任意
\begin_inset Formula $n+1$
\end_inset
个向量都线性相关, 因此向量组
\begin_inset Formula $E$
\end_inset
是
\begin_inset Formula $R^{n}$
\end_inset
的一个极大无关组, 且
\begin_inset Formula $\RR^{n}$
\end_inset
的秩等于
\begin_inset Formula $n$
\end_inset
.
\end_layout
\end_deeper
\begin_layout Subsection
向量组的秩
\end_layout
\begin_layout Frame
\begin_inset Argument 4
status open
\begin_layout Plain Layout
向量组的秩
\end_layout
\end_inset
\end_layout
\begin_deeper
\begin_layout Definition
向量组
\begin_inset Formula $\alpha_{1},\alpha_{2},\cdots,\alpha_{s}$
\end_inset
的极大无关组所含向量的个数称为该
\series bold
向量组的秩
\series default
, 记为
\end_layout
\begin_layout Definition
\begin_inset Formula
\[
r\left(\alpha_{1},\alpha_{2},\cdots,\alpha_{s}\right).
\]
\end_inset
\end_layout
\begin_layout Standard
规定: 只由零向量组成的向量组的秩为
\begin_inset Formula $0$
\end_inset
.
\end_layout
\end_deeper
\begin_layout Subsection
矩阵与向量组秩的关系
\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
矩阵的秩与向量组的秩
\end_layout
\end_inset
\end_layout
\begin_deeper
\begin_layout Theorem
\begin_inset Argument 1
status open
\begin_layout Plain Layout
p.
83, 定理 10
\end_layout
\end_inset
\begin_inset CommandInset label
LatexCommand label
name "thm:3.4-2"
\end_inset
设
\begin_inset Formula $A$
\end_inset
为
\begin_inset Formula $m\times n$
\end_inset
矩阵, 则矩阵
\begin_inset Formula $A$
\end_inset
的秩等于它的列向量组的秩, 也等于它的行向量组的秩.
\end_layout
\begin_layout Corollary
\begin_inset Argument 1
status open
\begin_layout Plain Layout
p.
83, 推论 7
\end_layout
\end_inset
矩阵
\begin_inset Formula $A$
\end_inset
的行向量组的秩与列向量组的秩相等.
\end_layout
\begin_layout Standard
由定理
\begin_inset CommandInset ref
LatexCommand ref
reference "thm:3.4-2"
plural "false"
caps "false"
noprefix "false"
\end_inset
知, 若
\begin_inset Formula $D_{r}$
\end_inset
是矩阵
\begin_inset Formula $A$
\end_inset
的一个最高阶非零子式, 则
\begin_inset Formula $D_{r}$
\end_inset
所在的
\begin_inset Formula $r$
\end_inset
列就是
\begin_inset Formula $A$
\end_inset
的列向量组的一个极大无关组;
\begin_inset Formula $D_{r}$
\end_inset
所在的
\begin_inset Formula $r$
\end_inset
行即是
\begin_inset Formula $A$
\end_inset
的行向量组的一个极大无关组.
\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
矩阵与向量组秩的关系
\end_layout
\end_inset
\end_layout
\begin_deeper
\begin_layout Standard
以向量组中各向量为列向量组成矩阵后, 只作初等行变换将该矩阵化为行阶梯形矩阵, 则可直接写出所求向量组的极大无关组;
\end_layout
\begin_layout Standard
同理, 也可
\uuline on
以向量组
\uuline default
中各向量为行向量组成矩阵, 通过作初等列变换来求所求向量组的极大无关组.
\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
向量组的秩
\end_layout
\end_inset
\end_layout
\begin_deeper
\begin_layout Example
求向量组
\begin_inset Formula
\[
\alpha_{1}=(1,2,-1,1)^{T},\ \alpha_{2}=(2,0,t,0)^{T},\ \alpha_{3}=(0,-4,5,-2)^{T},\ \alpha_{4}=(3,-2,t+4,-1)^{T}
\]
\end_inset
的秩和一个极大无关组.
\end_layout
\begin_layout Solution*
向量的分量中含参数
\begin_inset Formula $t$
\end_inset
, 向量组的秩和极大无关组与
\begin_inset Formula $t$
\end_inset
的取值有关.
对下列矩阵作初等行变换:
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
vspace{-4mm}
\end_layout
\end_inset
\end_layout
\begin_layout Solution*
\begin_inset Formula
\begin{align*}
\begin{bmatrix}\alpha_{1} & \alpha_{2} & \alpha_{3} & \alpha_{4}\end{bmatrix} & =\begin{bmatrix}1 & 2 & 0 & 3\\
2 & 0 & -4 & -2\\
-1 & t & 5 & t+4\\
1 & 0 & -2 & -1
\end{bmatrix}\\
& \longrightarrow\begin{bmatrix}1 & 2 & 0 & 3\\
0 & -4 & -4 & -8\\
0 & t+2 & 5 & t+7\\
0 & -2 & -2 & -4
\end{bmatrix}\longrightarrow\begin{bmatrix}1 & 2 & 0 & 3\\
0 & 1 & 1 & 2\\
0 & 0 & 3-t & 3-t\\
0 & 0 & 0 & 0
\end{bmatrix}
\end{align*}
\end_inset
显然,
\begin_inset Formula $\alpha_{1},\alpha_{2}$
\end_inset
线性无关, 且
\end_layout
\begin_layout Solution*
(1)
\begin_inset Formula $t=3$
\end_inset
时, 则
\begin_inset Formula $r\left(\alpha_{1},\alpha_{2},\alpha_{3},\alpha_{4}\right)=2$
\end_inset
, 且
\begin_inset Formula $\alpha_{1},\alpha_{2}$
\end_inset
是极大无关组;
\end_layout
\begin_layout Solution*
(2)
\begin_inset Formula $t\neq3$
\end_inset
时, 则
\begin_inset Formula $r\left(\alpha_{1},\alpha_{2},\alpha_{3},\alpha_{4}\right)=3$
\end_inset
, 且
\begin_inset Formula $\alpha_{1},\alpha_{2},\alpha_{3}$
\end_inset
是极大无关组;
\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
极大线性无关向量组
\end_layout
\end_inset
\end_layout
\begin_deeper
\begin_layout Example
设矩阵
\begin_inset Formula $A=\begin{bmatrix}2 & -1 & -1 & 1 & 2\\
1 & 1 & -2 & 1 & 4\\
4 & -6 & 2 & -2 & 4\\
3 & 6 & -9 & 7 & 9
\end{bmatrix}$
\end_inset
, 求矩阵
\begin_inset Formula $A$
\end_inset
的列向量组的一个极大无关组并把不属于极大无关组的列向量用极大无关组线性表示.
\end_layout
\begin_layout Solution*
对
\begin_inset Formula $A$
\end_inset
施行初等变换化为行阶梯形矩阵:
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
vspace{-4mm}
\end_layout
\end_inset
\begin_inset Formula
\begin{equation}
A\longrightarrow\begin{bmatrix}1 & 1 & -2 & 1 & 4\\
0 & 1 & -1 & 1 & 0\\
0 & 0 & 0 & 1 & -3\\
0 & 0 & 0 & 0 & 0
\end{bmatrix},\label{eq:3.4-1}
\end{equation}
\end_inset
知
\begin_inset Formula $r(A)=3$
\end_inset
, 故列向量组的极大无关组含
\begin_inset Formula $3$
\end_inset
个向量.
\end_layout
\begin_layout Solution*
而三个非零首元在第
\begin_inset Formula $1,2,4$
\end_inset
三列, 故
\begin_inset Formula $\alpha_{1},\alpha_{2},\alpha_{4}$
\end_inset
为列向量组的一个极大无关组.
也即
\begin_inset Formula $r\left(\alpha_{1},\alpha_{2},\alpha_{4}\right)=3$
\end_inset
, 故
\begin_inset Formula $\alpha_{1},\alpha_{2},\alpha_{4}$
\end_inset
线性无关.
\end_layout
\begin_layout Standard
\begin_inset Separator plain
\end_inset
\end_layout
\begin_layout Solution*
将
\begin_inset Formula $A$
\end_inset
化为行最简形矩阵, (将三个非零首元所在列清零, 与 (
\begin_inset CommandInset ref
LatexCommand ref
reference "eq:3.4-1"
plural "false"
caps "false"
noprefix "false"
\end_inset
) 式比较):
\end_layout
\begin_layout Solution*
\begin_inset ERT
status open
\begin_layout Plain Layout
$$
\end_layout
\begin_layout Plain Layout
A
\backslash
longrightarrow
\backslash
begin{bNiceMatrix}[first-row]
\end_layout
\begin_layout Plain Layout
\backslash
alpha_{1}&
\backslash
alpha_{2}&
\backslash
alpha_{3}&
\backslash
alpha_{4}&
\backslash
alpha_{5}
\backslash
\backslash
1&0&-1&0&4
\backslash
\backslash
0&1&-1&0&3
\backslash
\backslash
0&0&0&1&-3
\backslash
\backslash
0&0&0&0&0
\end_layout
\begin_layout Plain Layout
\backslash
end{bNiceMatrix},
\end_layout
\begin_layout Plain Layout
$$
\end_layout
\end_inset
于是, 可以看出
\begin_inset Formula $\begin{cases}
\alpha_{3}=-\alpha_{1}-\alpha_{2}\\
\alpha_{5}=4\alpha_{1}+3\alpha_{2}-3\alpha_{4}
\end{cases}$
\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
矩阵与向量组秩的关系
\end_layout
\end_inset
\end_layout
\begin_deeper
\begin_layout Theorem
\begin_inset Argument 1
status open
\begin_layout Plain Layout
向量间线性关系的判定定理
\end_layout
\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
\begin_inset Formula $r(B)\leq r(A)$
\end_inset
.
\end_layout
\begin_deeper
\begin_layout Pause
\end_layout
\end_deeper
\begin_layout Corollary
\begin_inset CommandInset label
LatexCommand label
name "cor:3.4-1"
\end_inset
等价的向量组的秩相等.
\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
矩阵与向量组秩的关系
\end_layout
\end_inset
\end_layout
\begin_deeper
\begin_layout Corollary
\begin_inset CommandInset label
LatexCommand label
name "cor:3.4-2"
\end_inset
设
\begin_inset Formula $C_{m\times n}=A_{m\times s}B_{s\times n}$
\end_inset
, 则
\begin_inset Formula $r(C)\leq\min\{r(A),r(B)\}$
\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 Corollary
\begin_inset CommandInset label
LatexCommand label
name "cor:3.4-3"
\end_inset
设向量组
\begin_inset Formula $B$
\end_inset
是向量组
\begin_inset Formula $A$
\end_inset
的部分组, 若向量组
\begin_inset Formula $B$
\end_inset
线性无关, 且向量组
\begin_inset Formula $A$
\end_inset
能由向量组
\begin_inset Formula $B$
\end_inset
线性表示, 则向量组
\begin_inset Formula $B$
\end_inset
是向量组
\begin_inset Formula $A$
\end_inset
的一个极大无关组.
\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
秩不等式
\end_layout
\end_inset
\end_layout
\begin_deeper
\begin_layout Example
设
\begin_inset Formula $A_{m\times n}$
\end_inset
及
\begin_inset Formula $B_{n\times s}$
\end_inset
为两个矩阵, 证明:
\begin_inset Formula $A$
\end_inset
与
\begin_inset Formula $B$
\end_inset
乘积的秩不大于
\begin_inset Formula $A$
\end_inset
的秩和
\begin_inset Formula $B$
\end_inset
的秩, 即
\begin_inset Formula $r(AB)\leq\min(r(A),r(B))$
\end_inset
.
\end_layout
\begin_layout Proof
\series bold
将
\begin_inset Formula $A$
\end_inset
看成列向量组成的行矩阵
\series default
:
\begin_inset Formula $A=\left(a_{ij}\right)_{m\times n}=\left(\alpha_{1},\alpha_{2},\cdots,\alpha_{n}\right)$
\end_inset
, 设
\begin_inset Formula $B=\left(b_{ij}\right)_{n\times s}$
\end_inset
,
\begin_inset Formula $AB=C=\left(c_{ij}\right)_{m\times s}=\left(\gamma_{1},\gamma_{2},\cdots,\gamma_{s}\right)$
\end_inset
, 即
\begin_inset Formula
\[
\left(\gamma_{1},\gamma_{2},\cdots,\gamma_{s}\right)=\left(\alpha_{1},\alpha_{2},\cdots,\alpha_{n}\right)\begin{bmatrix}b_{11} & \cdots & b_{1j} & \cdots & b_{1s}\\
b_{21} & \cdots & b_{2j} & \cdots & b_{2s}\\
\vdots & \ddots & \vdots & \ddots & \vdots\\
b_{n1} & \cdots & b_{nj} & \cdots & b_{ns}
\end{bmatrix}
\]
\end_inset
因此有
\begin_inset Formula $\gamma_{j}=b_{1j}\alpha_{1}+b_{2j}\alpha_{2}+\cdots+b_{nj}\alpha_{n}$
\end_inset
, (
\begin_inset Formula $j=1,2,\cdots,s$
\end_inset
), 即
\begin_inset Formula $AB$
\end_inset
的列向量组
\begin_inset Formula $\gamma_{1},\gamma_{2},\cdots,\gamma_{s}$
\end_inset
可由
\begin_inset Formula $A$
\end_inset
的列向量组
\begin_inset Formula $\alpha_{1},\alpha_{2},\cdots,\alpha_{n}$
\end_inset
线性表示.
\end_layout
\begin_layout Standard
\begin_inset Separator plain
\end_inset
\end_layout
\begin_layout Proof
故
\begin_inset Formula $\gamma_{1},\gamma_{2},\cdots,\gamma_{s}$
\end_inset
的极大无关组可由
\begin_inset Formula $\alpha_{1},\alpha_{2},\cdots,\alpha_{n}$
\end_inset
的极大无关组线性表示, 由
\series bold
向量间线性关系的判定定理
\series default
:
\begin_inset Formula
\[
r(AB)\leq r(A).
\]
\end_inset
\end_layout
\begin_layout Proof
类似地:
\series bold
将
\begin_inset Formula $B$
\end_inset
看成行向量的列矩阵
\series default
,
\begin_inset Formula $B=\left(b_{ij}\right)=\begin{bmatrix}\beta_{1}\\
\beta_{2}\\
\vdots\\
\beta_{n}
\end{bmatrix}$
\end_inset
, 可以证明:
\begin_inset Formula $r(AB)\leq r(B)$
\end_inset
.
因此,
\begin_inset Formula $r(AB)\leq\min(r(A),r(B))$
\end_inset
.
\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 3
status open
\begin_layout Plain Layout
allowframebreaks
\end_layout
\end_inset
\begin_inset Argument 4
status open
\begin_layout Plain Layout
向量组的等价
\end_layout
\end_inset
\end_layout
\begin_deeper
\begin_layout Example
设向量组
\begin_inset Formula $B$
\end_inset
能由向量组
\begin_inset Formula $A$
\end_inset
线性表示, 且它们的秩相等, 证明向量组
\begin_inset Formula $A$
\end_inset
与向量组
\begin_inset Formula $B$
\end_inset
等价.
\end_layout
\begin_layout Proof
\series bold
证法一:
\series default
只要证明向量组
\begin_inset Formula $A$
\end_inset
能由向量组
\begin_inset Formula $B$
\end_inset
线性表示.
设两个向量组的秩都为
\begin_inset Formula $s$
\end_inset
, 并设
\begin_inset Formula $A$
\end_inset
组和
\begin_inset Formula $B$
\end_inset
组的极大无关组依次为
\begin_inset Formula $A_{0}:\alpha_{1},\cdots,\alpha_{s}$
\end_inset
和
\begin_inset Formula $B_{0}:\beta_{1},\cdots,\beta_{s}$
\end_inset
, 因
\begin_inset Formula $B$
\end_inset
组能由
\begin_inset Formula $A$
\end_inset
组线性表示, 故
\begin_inset Formula $B_{0}$
\end_inset
组能由
\begin_inset Formula $A_{0}$
\end_inset
组线性表示, 即有
\begin_inset Formula $s$
\end_inset
阶方阵
\begin_inset Formula $K_{s}$
\end_inset
使
\begin_inset Formula $\left(\beta_{1},\cdots,\beta_{s}\right)=\left(\alpha_{1},\cdots,\alpha_{s}\right)K_{s}$
\end_inset
.
因
\begin_inset Formula $B_{0}$
\end_inset
组线性无关, 故由推论
\begin_inset CommandInset ref
LatexCommand ref
reference "cor:3.4-2"
plural "false"
caps "false"
noprefix "false"
\end_inset
知
\begin_inset Formula
\[
r\left(\beta_{1},\cdots,\beta_{s}\right)=s\Longrightarrow r\left(K_{s}\right)\ge r(A_{0}K_{s})=r\left(\beta_{1},\cdots,\beta_{s}\right)=s.
\]
\end_inset
但
\begin_inset Formula $r\left(K_{s}\right)\leq s$
\end_inset
, 因此
\begin_inset Formula $r\left(K_{s}\right)=s$
\end_inset
.
于是矩阵
\begin_inset Formula $K_{s}$
\end_inset
可逆, 并有
\begin_inset Formula $\left(\alpha_{1},\cdots,\alpha_{s}\right)=\left(\beta_{1},\cdots,\beta_{s}\right)K_{s}^{-1}$
\end_inset
, 即
\begin_inset Formula $A_{0}$
\end_inset
组能由
\begin_inset Formula $B_{0}$
\end_inset
组线性表示, 从而
\begin_inset Formula $A$
\end_inset
组能由
\begin_inset Formula $B$
\end_inset
组线性表示.
\end_layout
\begin_layout Standard
\begin_inset Separator plain
\end_inset
\end_layout
\begin_layout Proof
\series bold
证法二:
\series default
设向量组
\begin_inset Formula $A$
\end_inset
和
\begin_inset Formula $B$
\end_inset
的秩都有为
\begin_inset Formula $s$
\end_inset
.
因
\begin_inset Formula $B$
\end_inset
组能由
\begin_inset Formula $A$
\end_inset
组线性表示, 故
\begin_inset Formula $A$
\end_inset
组和
\begin_inset Formula $B$
\end_inset
组合并而成的向量组
\begin_inset Formula $(A,B)$
\end_inset
能由
\begin_inset Formula $A$
\end_inset
组线性表示.
而
\begin_inset Formula $A$
\end_inset
组是
\begin_inset Formula $(A,B)$
\end_inset
组的部分组, 故
\begin_inset Formula $A$
\end_inset
组总能由
\begin_inset Formula $(A,B)$
\end_inset
组线性表示.
所以
\begin_inset Formula $(A,B)$
\end_inset
组与
\begin_inset Formula $A$
\end_inset
组等价, 因此
\begin_inset Formula $(A,B)$
\end_inset
组的秩也为
\begin_inset Formula $s$
\end_inset
(见推论
\begin_inset CommandInset ref
LatexCommand ref
reference "cor:3.4-1"
plural "false"
caps "false"
noprefix "false"
\end_inset
).
\end_layout
\begin_layout Proof
又因
\begin_inset Formula $B$
\end_inset
组的秩也为
\begin_inset Formula $s$
\end_inset
, 故
\begin_inset Formula $B$
\end_inset
组的极大无关组
\begin_inset Formula $B_{0}$
\end_inset
含
\begin_inset Formula $s$
\end_inset
个向量; 由于向量组
\begin_inset Formula $B$
\end_inset
能由向量组
\begin_inset Formula $A$
\end_inset
线性表示, 因此
\begin_inset Formula $(A,B)$
\end_inset
的极大无关组的秩与向量组
\begin_inset Formula $A$
\end_inset
的秩相等, 并等于向量组
\begin_inset Formula $B$
\end_inset
(或者
\begin_inset Formula $B_{0}$
\end_inset
) 的秩
\begin_inset Formula $s$
\end_inset
, 因此
\begin_inset Formula $B_{0}$
\end_inset
组也是
\begin_inset Formula $(A,B)$
\end_inset
组的极大无关组, 从而
\begin_inset Formula $(A,B)$
\end_inset
组与
\begin_inset Formula $B_{0}$
\end_inset
组等价, 由
\begin_inset Formula $A$
\end_inset
组与
\begin_inset Formula $(A,B)$
\end_inset
组等价,
\begin_inset Formula $(A,B)$
\end_inset
与
\begin_inset Formula $B_{0}$
\end_inset
等价, 推知
\begin_inset Formula $A$
\end_inset
组与
\begin_inset Formula $B_{0}$
\end_inset
组等价.
\end_layout
\begin_layout Remark*
本例把证明两向量组
\begin_inset Formula $A$
\end_inset
与
\begin_inset Formula $B$
\end_inset
等价, 转换为证明它们的极大无关组
\begin_inset Formula $A_{0}$
\end_inset
与
\begin_inset Formula $B_{0}$
\end_inset
等到价.
\end_layout
\begin_layout Remark*
证法一证明
\begin_inset Formula $B_{0}$
\end_inset
用
\begin_inset Formula $A_{0}$
\end_inset
线性表示的系数矩阵可逆; 证法二实质上是证明
\begin_inset Formula $A_{0}$
\end_inset
与
\begin_inset Formula $B_{0}$
\end_inset
都是向量组
\begin_inset Formula $(A,B)$
\end_inset
的极大无关组.
\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
向量组的等价
\end_layout
\end_inset
\end_layout
\begin_deeper
\begin_layout Example
已知
\begin_inset Formula $\left(\alpha_{1},\alpha_{2}\right)=\begin{bmatrix}2 & 3\\
0 & -2\\
-1 & 1\\
3 & -1
\end{bmatrix}$
\end_inset
,
\begin_inset Formula $\left(\beta_{1},\beta_{2}\right)=\begin{bmatrix}-5 & 4\\
6 & -4\\
-5 & 3\\
9 & -5
\end{bmatrix}$
\end_inset
, 证明向量组
\begin_inset Formula $\left(\alpha_{1},\alpha_{2}\right)$
\end_inset
与
\begin_inset Formula $\left(\beta_{1},\beta_{2}\right)$
\end_inset
等价.
\end_layout
\begin_layout Proof
要证存在
\begin_inset Formula $2$
\end_inset
阶方阵
\begin_inset Formula $X,Y$
\end_inset
, 使
\begin_inset Formula $\left(\beta_{1},\beta_{2}\right)=\left(\alpha_{1},\alpha_{2}\right)X$
\end_inset
,
\begin_inset Formula $\left(\alpha_{1},\alpha_{2}\right)=\left(\beta_{1},\beta_{2}\right)Y$
\end_inset
.
\end_layout
\begin_layout Proof
先求
\begin_inset Formula $X$
\end_inset
.
对增广矩阵
\begin_inset Formula $\left(\alpha_{1},\alpha_{2},\beta_{1},\beta_{2}\right)$
\end_inset
施行初等行变换:
\begin_inset Formula
\begin{align*}
\left(\alpha_{1},\alpha_{2},\beta_{1},\beta_{2}\right) & =\begin{bmatrix}2 & 3 & -5 & 4\\
0 & -2 & 6 & -4\\
-1 & 1 & -5 & 3\\
3 & -1 & 9 & -5
\end{bmatrix}\longrightarrow\begin{bmatrix}-1 & 1 & -5 & 3\\
0 & -2 & 6 & -4\\
2 & 3 & -5 & 4\\
3 & -1 & 9 & -5
\end{bmatrix}\longrightarrow\begin{bmatrix}-1 & 1 & -5 & 3\\
0 & -2 & 6 & -4\\
0 & 5 & -15 & 10\\
0 & 2 & -6 & 4
\end{bmatrix}\\
& \longrightarrow\begin{bmatrix}-1 & 1 & -5 & 3\\
0 & 1 & -3 & 2\\
0 & 0 & 0 & 0\\
0 & 0 & 0 & 0
\end{bmatrix}\longrightarrow\begin{bmatrix}1 & 0 & 2 & -1\\
0 & 1 & -3 & 2\\
0 & 0 & 0 & 0\\
0 & 0 & 0 & 0
\end{bmatrix}\Longrightarrow X=\begin{bmatrix}2 & -1\\
-3 & 2
\end{bmatrix}
\end{align*}
\end_inset
因
\begin_inset Formula $|X|=1\neq0$
\end_inset
, 知
\begin_inset Formula $X$
\end_inset
可逆, 取
\begin_inset Formula $Y=X^{-1}$
\end_inset
, 即为所求.
因此向量组
\begin_inset Formula $\left(\alpha_{1},\alpha_{2}\right)$
\end_inset
与
\begin_inset Formula $\left(\beta_{1},\beta_{2}\right)$
\end_inset
等价.
\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
求向量组
\begin_inset Formula
\[
\alpha_{1}=(2,4,2)^{T},\ \alpha_{2}=(1,1,0)^{T},\ \alpha_{3}=(2,3,1)^{T},\ \alpha_{4}=(3,5,2)^{T}
\]
\end_inset
的一个极大无关组, 并把其余向量用该极大无关组线性表示.
\end_layout
\end_deeper
\begin_layout Frame
\end_layout
\end_body
\end_document
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