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\begin_body
\begin_layout Section
向量组的线性相关性
\end_layout
\begin_layout Subsection
线性相关性概念
\end_layout
\begin_layout Frame
\begin_inset Argument 4
status open
\begin_layout Plain Layout
线性相关与线性无关
\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
, 如果存在不全为零的数
\begin_inset Formula $k_{1},k_{2},\cdots,k_{s}$
\end_inset
, 使
\begin_inset Formula
\begin{equation}
k_{1}\alpha_{1}+k_{2}\alpha_{2}+\cdots+k_{s}\alpha_{s}=0,\label{eq:3.3-1}
\end{equation}
\end_inset
则称
\series bold
向量组
\begin_inset Formula $A$
\end_inset
线性相关
\series default
, 否则称为
\series bold
线性无关
\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
线性相关与线性无关
\end_layout
\end_inset
\end_layout
\begin_deeper
\begin_layout Remark*
(1) 当且仅当
\begin_inset Formula $k_{1}=k_{2}=\cdots=k_{s}=0$
\end_inset
时, (
\begin_inset CommandInset ref
LatexCommand ref
reference "eq:3.3-1"
plural "false"
caps "false"
noprefix "false"
\end_inset
) 式成立, 向量组
\begin_inset Formula $\alpha_{1},\alpha_{2},\cdots,\alpha_{s}$
\end_inset
线性无关;
\end_layout
\begin_layout Remark*
(2) 包含零向量的任何向量组是线性相关的;
\end_layout
\begin_layout Remark*
(3) 向量组只含有一个向量
\begin_inset Formula $\alpha$
\end_inset
时, 则
\end_layout
\begin_deeper
\begin_layout Itemize
\begin_inset Formula $\alpha\neq0$
\end_inset
的充分必要条件是
\begin_inset Formula $\alpha$
\end_inset
是线性无关的;
\end_layout
\begin_layout Itemize
\begin_inset Formula $\alpha=0$
\end_inset
的充分必要条件是
\begin_inset Formula $\alpha$
\end_inset
是线性相关的;
\end_layout
\end_deeper
\begin_layout Remark*
(4) 仅含两个向量的向量组线性相关的充分必要条件是这两个向量的对应分量成比例; 反之, 仅含两个向量的向量组线性无关的充分必要条件是这两个向量的对应分量不成比
例.
\end_layout
\begin_layout Remark*
(5) 两个向量线性相关的几何意义是这两个向量共线, 三个向量线性相关的几何意义是这三个向量共面.
\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
线性相关性概念
\end_layout
\end_inset
\end_layout
\begin_deeper
\begin_layout Example
设有
\begin_inset Formula $3$
\end_inset
个向量 (列向量):
\begin_inset Formula
\[
\alpha_{1}=\begin{bmatrix}1\\
0\\
1
\end{bmatrix},\quad\alpha_{2}=\begin{bmatrix}-1\\
2\\
2
\end{bmatrix},\quad\alpha_{3}=\begin{bmatrix}1\\
2\\
4
\end{bmatrix},
\]
\end_inset
不难验证
\begin_inset Formula $2\alpha_{1}+\alpha_{2}-\alpha_{3}=0$
\end_inset
, 因此
\begin_inset Formula $\alpha_{1},\alpha_{2},\alpha_{3}$
\end_inset
是
\begin_inset Formula $3$
\end_inset
个线性相关的
\begin_inset Formula $3$
\end_inset
维向量.
\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
线性相关性概念
\end_layout
\end_inset
\end_layout
\begin_deeper
\begin_layout Example
设有二个
\begin_inset Formula $2$
\end_inset
维向量:
\begin_inset Formula $e_{1}=\begin{bmatrix}1\\
0
\end{bmatrix}$
\end_inset
,
\begin_inset Formula $e_{2}=\begin{bmatrix}0\\
1
\end{bmatrix}$
\end_inset
, 如果它们线性相关, 那么存在不全为零的数
\begin_inset Formula $\lambda_{1},\lambda_{2}$
\end_inset
, 使
\begin_inset Formula
\[
\lambda_{1}e_{1}+\lambda_{2}e_{2}=0,
\]
\end_inset
也就是
\begin_inset Formula
\[
\lambda_{1}\begin{bmatrix}1\\
0
\end{bmatrix}+\lambda_{2}\begin{bmatrix}0\\
1
\end{bmatrix}=0,
\]
\end_inset
即
\begin_inset Formula
\[
\begin{bmatrix}\lambda_{1}\\
0
\end{bmatrix}+\begin{bmatrix}0\\
\lambda_{2}
\end{bmatrix}=\begin{bmatrix}\lambda_{1}\\
\lambda_{2}
\end{bmatrix}=0,
\]
\end_inset
于是
\begin_inset Formula $\lambda_{1}=0$
\end_inset
,
\begin_inset Formula $\lambda_{2}=0$
\end_inset
, 这同
\begin_inset Formula $\lambda_{1},\lambda_{2}$
\end_inset
不全为零的假定矛盾.
因此
\begin_inset Formula $e_{1},e_{2}$
\end_inset
是线性无关的两个向量.
\end_layout
\end_deeper
\begin_layout Subsection
线性相关性的判定
\end_layout
\begin_layout Frame
\begin_inset Argument 4
status open
\begin_layout Plain Layout
线性相关性的判定
\end_layout
\end_inset
\end_layout
\begin_deeper
\begin_layout Theorem
\begin_inset Argument 1
status open
\begin_layout Plain Layout
p.
74, 定理 1
\end_layout
\end_inset
向量组
\begin_inset Formula $\alpha_{1},\alpha_{2},\cdots,\alpha_{s}$
\end_inset
(
\begin_inset Formula $s\geq2$
\end_inset
) 线性相关的充分必要条件是向量组中至少有一个向量可由其余
\begin_inset Formula $s-1$
\end_inset
个向量线性表示.
\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
线性相关性的判定
\end_layout
\end_inset
\end_layout
\begin_deeper
\begin_layout Theorem
\begin_inset CommandInset label
LatexCommand label
name "thm:3.3-2"
\end_inset
设有列向量组
\begin_inset Formula $\alpha_{j}=\begin{bmatrix}a_{1j}\\
a_{2j}\\
\vdots\\
a_{nj}
\end{bmatrix}$
\end_inset
, (
\begin_inset Formula $j=1,2,\cdots,s$
\end_inset
), 则向量组
\begin_inset Formula $\alpha_{1},\alpha_{2},\cdots,\alpha_{s}$
\end_inset
线性相关的充要条件是: 矩阵
\begin_inset Formula $A=\left(\alpha_{1},\alpha_{2},\cdots,\alpha_{s}\right)$
\end_inset
的秩小于向量的个数
\begin_inset Formula $s$
\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
线性相关性的判定
\end_layout
\end_inset
\end_layout
\begin_deeper
\begin_layout Corollary
\begin_inset CommandInset label
LatexCommand label
name "cor:3.3-2"
\end_inset
\begin_inset Formula $n$
\end_inset
个
\begin_inset Formula $n$
\end_inset
维列向量组
\begin_inset Formula $\alpha_{1},\alpha_{2},\cdots,\alpha_{n}$
\end_inset
线性无关 (线性相关) 的充要条件是: 矩阵
\begin_inset Formula $A=\left(\alpha_{1},\alpha_{2},\cdots,\alpha_{n}\right)$
\end_inset
的秩等于 (小于) 向量的个数
\begin_inset Formula $n$
\end_inset
.
\end_layout
\begin_layout Standard
\begin_inset Separator plain
\end_inset
\end_layout
\begin_layout Corollary
\begin_inset Argument 1
status open
\begin_layout Plain Layout
定理 6, 推论 2
\end_layout
\end_inset
\begin_inset CommandInset label
LatexCommand label
name "cor:3.3-3"
\end_inset
\begin_inset Formula $n$
\end_inset
个
\begin_inset Formula $n$
\end_inset
维列向量组
\begin_inset Formula $\alpha_{1},\alpha_{2},\cdots,\alpha_{n}$
\end_inset
线性无关 (线性相关) 的充要条件是: 矩阵
\begin_inset Formula $A=\left(\alpha_{1},\alpha_{2},\cdots,\alpha_{n}\right)$
\end_inset
的行列式不等于 (等于) 零.
\end_layout
\begin_layout Standard
\begin_inset Separator plain
\end_inset
\end_layout
\begin_layout Remark*
上述结论对于矩阵的行向量组也同样成立.
\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
用矩阵判定向量组的线性相关性
\end_layout
\end_inset
\end_layout
\begin_deeper
\begin_layout Example
\begin_inset Formula $n$
\end_inset
维向量组
\begin_inset Formula
\[
\varepsilon_{1}=(1,0,\cdots,0)^{T},\ \varepsilon_{2}=(0,1,\cdots,0)^{T},\ \cdots,\ \varepsilon_{n}=(0,0,\cdots,1)^{T}.
\]
\end_inset
称为
\series bold
\begin_inset Formula $n$
\end_inset
维单位向量组
\series default
, 讨论其线性相关性.
\end_layout
\begin_deeper
\begin_layout Pause
\end_layout
\end_deeper
\begin_layout Solution*
\begin_inset Formula $n$
\end_inset
维单位坐标向量组构成的矩阵
\begin_inset Formula
\[
E=\left(\varepsilon_{1},\varepsilon_{2},\cdots,\varepsilon_{n}\right)=\begin{bmatrix}1 & 0 & \cdots & 0\\
0 & 1 & \cdots & 0\\
\vdots & \vdots & \ddots & \vdots\\
0 & 0 & \cdots & 1
\end{bmatrix}
\]
\end_inset
是
\begin_inset Formula $n$
\end_inset
阶单位矩阵.
\end_layout
\begin_layout Solution*
由
\begin_inset Formula $|E|=1\neq0$
\end_inset
, 知
\begin_inset Formula $r(E)=n$
\end_inset
.
即
\begin_inset Formula $r(E)$
\end_inset
等于向量组中向量的个数, 故由推论
\begin_inset CommandInset ref
LatexCommand ref
reference "cor:3.3-2"
plural "false"
caps "false"
noprefix "false"
\end_inset
或推论
\begin_inset CommandInset ref
LatexCommand ref
reference "cor:3.3-3"
plural "false"
caps "false"
noprefix "false"
\end_inset
知此向量是线性无关的.
\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
线性相关性的判定
\end_layout
\end_inset
\end_layout
\begin_deeper
\begin_layout Example
已知
\begin_inset Formula $\alpha_{1}=\begin{bmatrix}1\\
1\\
1
\end{bmatrix}$
\end_inset
,
\begin_inset Formula $\alpha_{2}=\begin{bmatrix}0\\
2\\
5
\end{bmatrix}$
\end_inset
,
\begin_inset Formula $\alpha_{3}=\begin{bmatrix}2\\
4\\
7
\end{bmatrix}$
\end_inset
, 试讨论向量组
\begin_inset Formula $\alpha_{1},\alpha_{2},\alpha_{3}$
\end_inset
及
\begin_inset Formula $\alpha_{1},\alpha_{2}$
\end_inset
的线性相关性.
\end_layout
\begin_layout Solution*
对矩阵
\begin_inset Formula $A=\left(\alpha_{1},\alpha_{2},\alpha_{3}\right)$
\end_inset
施行初等行变换成行阶梯形矩阵, 可同时看出矩阵
\begin_inset Formula $A$
\end_inset
及
\begin_inset Formula $B=\left(\alpha_{1},\alpha_{2}\right)$
\end_inset
的秩, 利用定理
\begin_inset CommandInset ref
LatexCommand ref
reference "thm:3.3-2"
plural "false"
caps "false"
noprefix "false"
\end_inset
即可得出结论.
\begin_inset Formula
\[
\left(\alpha_{1},\alpha_{2},\alpha_{3}\right)=\begin{bmatrix}1 & 0 & 2\\
1 & 2 & 4\\
1 & 5 & 7
\end{bmatrix}\xrightarrow[r_{3}-r_{1}]{r_{2}-r_{1}}\begin{bmatrix}1 & 0 & 2\\
0 & 2 & 2\\
0 & 5 & 5
\end{bmatrix}\xrightarrow{r_{1}-\frac{5}{2}r_{2}}\begin{bmatrix}1 & 0 & 2\\
0 & 2 & 2\\
0 & 0 & 0
\end{bmatrix},
\]
\end_inset
易见,
\begin_inset Formula $r(A)=2$
\end_inset
,
\begin_inset Formula $r(B)=2$
\end_inset
, 故向量组
\begin_inset Formula $\alpha_{1},\alpha_{2},\alpha_{3}$
\end_inset
线性相关.
向量组
\begin_inset Formula $\alpha_{1},\alpha_{2}$
\end_inset
线性无关.
\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
线性相关性的判定
\end_layout
\end_inset
\end_layout
\begin_deeper
\begin_layout Corollary
\begin_inset Argument 1
status open
\begin_layout Plain Layout
定理 7
\end_layout
\end_inset
当向量组中所含向量的个数大于向量的维数时, 此向量组必线性相关.
\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
线性相关性的判定
\end_layout
\end_inset
\end_layout
\begin_deeper
\begin_layout Theorem
\begin_inset Argument 1
status open
\begin_layout Plain Layout
定理 3
\end_layout
\end_inset
如果向量组中有一部分向量 (部分组) 线性相关, 则整个向量组线性相关.
\end_layout
\begin_layout Corollary
线性无关的向量组中的任何部分组都线性无关.
\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
线性相关性与向量的线性表示
\end_layout
\end_inset
\end_layout
\begin_deeper
\begin_layout Theorem
\begin_inset Argument 1
status open
\begin_layout Plain Layout
p.
75, 定理 2
\end_layout
\end_inset
\begin_inset CommandInset label
LatexCommand label
name "thm:2.13"
\end_inset
若向量组
\begin_inset Formula $\alpha_{1},\cdots,\alpha_{s},\beta$
\end_inset
线性相关, 而向量组
\begin_inset Formula $\alpha_{1},\alpha_{2},\cdots,\alpha_{s}$
\end_inset
线性无关, 则向量
\begin_inset Formula $\beta$
\end_inset
可由
\begin_inset Formula $\alpha_{1},\alpha_{2},\cdots,\alpha_{s}$
\end_inset
线性表示且表示法唯一.
\end_layout
\begin_layout Standard
设实数
\begin_inset Formula $k_{1},k_{2},\cdots,k_{s},k$
\end_inset
使得
\begin_inset Formula
\[
k_{1}\alpha_{1}+k_{2}\alpha_{2}+\cdots+k_{s}\alpha_{s}+k\beta=0.
\]
\end_inset
\end_layout
\begin_layout Itemize
若
\begin_inset Formula $k=0$
\end_inset
, 则由向量组
\begin_inset Formula $\alpha_{1},\alpha_{2},\cdots,\alpha_{s}$
\end_inset
线性无关知,
\begin_inset Formula $k_{1},k_{2},\cdots,k_{s}$
\end_inset
均为零;
\end_layout
\begin_deeper
\begin_layout Itemize
因向量组
\begin_inset Formula $\alpha_{1},\cdots,\alpha_{s},\beta$
\end_inset
线性相关, 所以实数
\begin_inset Formula $k_{1},k_{2},\cdots,k_{s},k$
\end_inset
不能只有零解.
\end_layout
\end_deeper
\begin_layout Itemize
若
\begin_inset Formula $k\ne0$
\end_inset
, 则
\begin_inset Formula $\beta=-\frac{k_{1}}{k}\alpha_{1}-\frac{k_{2}}{k}\alpha_{2}-\cdots-\frac{k_{s}}{k}\alpha_{s}$
\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
线性相关性与向量的线性表示
\end_layout
\end_inset
\end_layout
\begin_deeper
\begin_layout Theorem
\begin_inset Argument 1
status open
\begin_layout Plain Layout
p.
78, 定理 8
\end_layout
\end_inset
设有两向量组
\end_layout
\begin_layout Theorem
\begin_inset Formula
\[
A:\alpha_{1},\alpha_{2},\cdots,\alpha_{s};\quad B:\beta_{1},\beta_{2},\cdots,\beta_{t},
\]
\end_inset
向量组
\begin_inset Formula $B$
\end_inset
能由向量组
\begin_inset Formula $A$
\end_inset
线性表示, 若
\begin_inset Formula $s<t$
\end_inset
, 则向量组
\begin_inset Formula $B$
\end_inset
线性相关.
\end_layout
\begin_deeper
\begin_layout Pause
\end_layout
\end_deeper
\begin_layout Corollary
\begin_inset Argument 1
status open
\begin_layout Plain Layout
p.
79, 推论 4
\end_layout
\end_inset
向量组
\begin_inset Formula $B$
\end_inset
能由向量组
\begin_inset Formula $A$
\end_inset
线性表示, 若向量组
\begin_inset Formula $B$
\end_inset
线性无关, 则
\begin_inset Formula $s\geq t$
\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
线性相关性与向量的线性表示
\end_layout
\end_inset
\end_layout
\begin_deeper
\begin_layout Corollary
\begin_inset CommandInset label
LatexCommand label
name "cor:3.2-6"
\end_inset
设向量组
\begin_inset Formula $A$
\end_inset
与
\begin_inset Formula $B$
\end_inset
可以相互线性表示, 若
\begin_inset Formula $A$
\end_inset
与
\begin_inset Formula $B$
\end_inset
都是线性无关的, 则
\begin_inset Formula $s=t$
\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
线性相关性的判定
\end_layout
\end_inset
\end_layout
\begin_deeper
\begin_layout Example
判断下列向量组是否线性相关:
\end_layout
\begin_layout Example
\begin_inset Formula
\[
\alpha_{1}=\begin{bmatrix}1\\
2\\
-1\\
5
\end{bmatrix},\quad\alpha_{2}=\begin{bmatrix}2\\
-1\\
1\\
1
\end{bmatrix},\quad\alpha_{3}=\begin{bmatrix}4\\
3\\
-1\\
11
\end{bmatrix}.
\]
\end_inset
\end_layout
\begin_layout Solution*
对矩阵
\begin_inset Formula $\left(\alpha_{1},\alpha_{2},\alpha_{3}\right)$
\end_inset
施以初等行变换化为阶梯形矩阵:
\begin_inset Formula
\[
\begin{bmatrix}1 & 2 & 4\\
2 & -1 & 3\\
-1 & 1 & -1\\
5 & 1 & 11
\end{bmatrix}\longrightarrow\begin{bmatrix}1 & 2 & 4\\
0 & -5 & -5\\
0 & 3 & 3\\
0 & -9 & -9
\end{bmatrix}\rightarrow\begin{bmatrix}1 & 2 & 4\\
0 & 1 & 1\\
0 & 0 & 0\\
0 & 0 & 0
\end{bmatrix},
\]
\end_inset
秩
\begin_inset Formula $\left(\alpha_{1},\alpha_{2},\alpha_{3}\right)=2<3$
\end_inset
, 所以向量组
\begin_inset Formula $\alpha_{1},\alpha_{2},\alpha_{3}$
\end_inset
线性相关.
\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
线性相关性的判定
\end_layout
\end_inset
\end_layout
\begin_deeper
\begin_layout Example
证明: 若向量组
\begin_inset Formula $\alpha,\beta,\gamma$
\end_inset
线性无关, 则向量组
\begin_inset Formula $\alpha+\beta,\beta+\gamma,\gamma+\alpha$
\end_inset
亦线性无关.
\end_layout
\begin_layout Proof
设有一组数
\begin_inset Formula $k_{1},k_{2},k_{3}$
\end_inset
, 使
\begin_inset Formula
\begin{equation}
k_{1}(\alpha+\beta)+k_{2}(\beta+\gamma)+k_{3}(\gamma+\alpha)=0\label{eq:3.3-1-1}
\end{equation}
\end_inset
成立, 整理得
\begin_inset Formula $\left(k_{1}+k_{3}\right)\alpha+\left(k_{1}+k_{2}\right)\beta+\left(k_{2}+k_{3}\right)\gamma=0$
\end_inset
.
\end_layout
\begin_layout Proof
由
\begin_inset Formula $\alpha,\beta,\gamma$
\end_inset
线性无关, 故
\begin_inset Formula
\begin{equation}
\begin{cases}
k_{1}+k_{3}=0\\
k_{1}+k_{2}=0\\
k_{2}+k_{3}=0
\end{cases}\label{eq:3.3-2}
\end{equation}
\end_inset
因为
\begin_inset Formula $\begin{vmatrix}1 & 0 & 1\\
1 & 1 & 0\\
0 & 1 & 1
\end{vmatrix}=2\neq0$
\end_inset
, 故方程组 (
\begin_inset CommandInset ref
LatexCommand ref
reference "eq:3.3-2"
plural "false"
caps "false"
noprefix "false"
\end_inset
) 仅有零解.
即只有
\begin_inset Formula $k_{1}=k_{2}=k_{3}=0$
\end_inset
时 (
\begin_inset CommandInset ref
LatexCommand ref
reference "eq:3.3-1-1"
plural "false"
caps "false"
noprefix "false"
\end_inset
) 式才成立.
\end_layout
\begin_layout Proof
因而向量组
\begin_inset Formula $\alpha+\beta,\beta+\gamma,\gamma+\alpha$
\end_inset
线性无关.
\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
线性相关性的判定
\end_layout
\end_inset
\end_layout
\begin_deeper
\begin_layout Example
设向量组
\begin_inset Formula $\alpha_{1},\alpha_{2},\alpha_{3}$
\end_inset
线性相关, 向量组
\begin_inset Formula $\alpha_{2},\alpha_{3},\alpha_{4}$
\end_inset
线性无关, 证明
\end_layout
\begin_layout Example
(1)
\begin_inset Formula $\alpha_{1}$
\end_inset
能由
\begin_inset Formula $\alpha_{2},\alpha_{3}$
\end_inset
线性表示;
\end_layout
\begin_layout Example
(2)
\begin_inset Formula $\alpha_{4}$
\end_inset
不能由
\begin_inset Formula $\alpha_{1},\alpha_{2},\alpha_{3}$
\end_inset
线性表示.
\end_layout
\begin_layout Proof
(1) 因
\begin_inset Formula $\alpha_{2},\alpha_{3},\alpha_{4}$
\end_inset
线性无关, 故
\begin_inset Formula $\alpha_{2},\alpha_{3}$
\end_inset
线性无关, 而
\begin_inset Formula $\alpha_{1},\alpha_{2},\alpha_{3}$
\end_inset
线性相关, 从而
\begin_inset Formula $\alpha_{1}$
\end_inset
能由
\begin_inset Formula $\alpha_{2},\alpha_{3}$
\end_inset
线性表示
\begin_inset Foot
status open
\begin_layout Plain Layout
书本 p.75, 定理 2, 或本课件定理
\begin_inset CommandInset ref
LatexCommand ref
reference "thm:2.13"
plural "false"
caps "false"
noprefix "false"
\end_inset
\end_layout
\end_inset
;
\end_layout
\begin_layout Proof
(2) 用反证法.
假设
\begin_inset Formula $\alpha_{4}$
\end_inset
能由
\begin_inset Formula $\alpha_{1},\alpha_{2},\alpha_{3}$
\end_inset
线性表示, 而由 (1) 知
\begin_inset Formula $\alpha_{1}$
\end_inset
能由
\begin_inset Formula $\alpha_{2},\alpha_{3}$
\end_inset
线性表示, 因此
\begin_inset Formula $\alpha_{4}$
\end_inset
能由
\begin_inset Formula $\alpha_{2},\alpha_{3}$
\end_inset
表示, 这与
\begin_inset Formula $\alpha_{2},\alpha_{3},\alpha_{4}$
\end_inset
线性无关矛盾.
证毕.
\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 3
status open
\begin_layout Plain Layout
allowframebreaks
\end_layout
\end_inset
\begin_inset Argument 4
status open
\begin_layout Plain Layout
作业
\end_layout
\end_inset
\end_layout
\begin_deeper
\begin_layout Problem
试证明:
\end_layout
\begin_layout Problem
(1) 一个向量
\begin_inset Formula $\alpha$
\end_inset
线性相关的充要条件是
\begin_inset Formula $\alpha=0$
\end_inset
;
\end_layout
\begin_layout Problem
(2) 一个向量
\begin_inset Formula $\alpha$
\end_inset
线性无关的充要条件是
\begin_inset Formula $\alpha\neq0$
\end_inset
;
\end_layout
\begin_layout Problem
(3) 两个向量
\begin_inset Formula $\alpha,\beta$
\end_inset
线性相关的充要条件是
\begin_inset Formula $\alpha=k\beta$
\end_inset
或者
\begin_inset Formula $\beta=k\alpha$
\end_inset
(两式不一定同时成立).
\end_layout
\begin_layout Standard
\begin_inset Separator plain
\end_inset
\end_layout
\begin_layout Problem
判断向量组
\begin_inset Formula
\[
\alpha_{1}=(1,2,0,1)^{T},\ \alpha_{2}=(1,3,0,-1)^{T},\ \alpha_{3}=(-1,-1,1,0)^{T}
\]
\end_inset
是否线性相关.
\end_layout
\begin_layout Standard
\begin_inset Separator plain
\end_inset
\end_layout
\begin_layout Problem
判断向量组
\begin_inset Formula
\[
\alpha_{1}=(1,2,-1,5)^{T},\ \alpha_{2}=(2,-1,1,1)^{T},\ \alpha_{3}=(4,3,-1,11)^{T}
\]
\end_inset
是否线性相关.
\end_layout
\begin_layout Standard
\begin_inset Separator plain
\end_inset
\end_layout
\begin_layout Problem
设向量组
\begin_inset Formula $\alpha_{1},\alpha_{2},\alpha_{3}$
\end_inset
线性无关, 又
\begin_inset Formula $\beta_{1}=\alpha_{1}+\alpha_{2}+2\alpha_{3}$
\end_inset
,
\begin_inset Formula $\beta_{3}=\alpha_{1}-\alpha_{2}$
\end_inset
,
\begin_inset Formula $\beta_{3}=\alpha_{1}+\alpha_{3}$
\end_inset
, 证明
\begin_inset Formula $\beta_{1},\beta_{2},\beta_{3}$
\end_inset
线性相关.
\end_layout
\end_deeper
\begin_layout Frame
\end_layout
\end_body
\end_document
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