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Latex:线性代数【大学常用公式】
吴文中
公式编辑器
:Ⅰ) 像码字一样
Latex
,复杂公式轻松编辑; Ⅱ) 大学、高中、初中、小学常用公式,一键模板。
Note:① 点击链接,想怎么修改就怎么修改;② 复制代码,Latex代码一键获取;
m行n列矩阵:
\mathop{{A}}\nolimits_{{m \times n}}={ \left[ {\begin{array}{*{20}{c}} {\mathop{{a}}\nolimits_{{11}}}&{\mathop{{a}}\nolimits_{{12}}}&{ \cdots }&{\mathop{{a}}\nolimits_{{1n}}}\\ {\mathop{{a}}\nolimits_{{21}}}&{\mathop{{a}}\nolimits_{{22}}}&{ \cdots }&{\mathop{{a}}\nolimits_{{2n}}}\\ { \vdots }&{ \vdots }&{ \ddots }&{ \vdots }\\ {\mathop{{a}}\nolimits_{{m1}}}&{\mathop{{a}}\nolimits_{{m2}}}&{ \cdots }&{\mathop{{a}}\nolimits_{{mn}}} \end{array}} \right] }={ \left[ {\mathop{{a}}\nolimits_{{ij}}} \right] }
n阶方阵:
{\mathop{{A}}\nolimits_{{n \times n}}={ \left[ {\begin{array}{*{20}{c}} {\mathop{{a}}\nolimits_{{11}}}&{\mathop{{a}}\nolimits_{{12}}}&{ \cdots }&{\mathop{{a}}\nolimits_{{1n}}}\\ {\mathop{{a}}\nolimits_{{21}}}&{\mathop{{a}}\nolimits_{{22}}}&{ \cdots }&{\mathop{{a}}\nolimits_{{2n}}}\\ { \vdots }&{ \vdots }&{ \ddots }&{ \vdots }\\ {\mathop{{a}}\nolimits_{{n1}}}&{\mathop{{a}}\nolimits_{{n2}}}&{ \cdots }&{\mathop{{a}}\nolimits_{{nn}}} \end{array}} \right] }}
矩阵乘法:
\begin{array}{*{20}{l}} {A={\mathop{{ \left[ {\mathop{{a}}\nolimits_{{ij}}} \right] }}\nolimits_{{m \times n}}},B=\mathop{{ \left[ {\mathop{{b}}\nolimits_{{ij}}} \right] }}\nolimits_{{n \times s}}}\\ {\mathop{{c}}\nolimits_{{ij}}=\mathop{ \sum }\limits_{{k=1}}^{{n}}\mathop{{a}}\nolimits_{{ik}}\mathop{{b}}\nolimits_{{kj}}}\\ {C=AB={\mathop{{ \left[ {\mathop{{c}}\nolimits_{{ij}}} \right] }}\nolimits_{{m \times s}}}={\mathop{{ \left[ {\mathop{ \sum }\limits_{{k=1}}^{{n}}\mathop{{a}}\nolimits_{{ik}}\mathop{{b}}\nolimits_{{kj}}} \right] }}\nolimits_{{m \times s}}}} \end{array}
对称矩阵和反对称矩阵:
\begin{array}{*{20}{l}} {A=\mathop{{A}}\nolimits^{{T}}}\\ {A=-\mathop{{A}}\nolimits^{{T}}} \end{array}
对角阵:
\begin{array}{*{20}{l}} {\text{若}n\text{阶}\text{对}\text{角}\text{阵}\text{对}\text{角}\text{线}\text{上}\text{的}\text{元}\text{素}\text{都}\text{等}\text{于}1}\\ {\mathop{{I}}\nolimits_{{3}}={ \left[ {\begin{array}{*{20}{c}} {1}&{0}&{0}\\ {0}&{1}&{0}\\ {0}&{0}&{1} \end{array}} \right] }} \end{array}
矩阵加法:
\mathop{{A}}\nolimits_{{m \times n}}+\mathop{{B}}\nolimits_{{m \times n}}={\mathop{{ \left[ {\mathop{{a}}\nolimits_{{ij}}} \right] }}\nolimits_{{m \times n}}}+\mathop{{ \left[ {\mathop{{b}}\nolimits_{{ij}}} \right] }}\nolimits_{{m \times n}}=\mathop{{ \left[ {\mathop{{a}}\nolimits_{{ij}}+\mathop{{b}}\nolimits_{{ij}}} \right] }}\nolimits_{{m \times n}}
零矩阵:
\begin{array}{*{20}{l}} {\text{矩}\text{阵}\text{的}\text{所}\text{有}\text{元}\text{素}\text{都}\text{为}0}\\ {O={ \left[ {\begin{array}{*{20}{c}} {0}&{0}&{ \cdots }&{0}\\ {0}&{0}&{ \cdots }&{0}\\ { \vdots }&{ \vdots }&{ \ddots }&{ \vdots }\\ {0}&{0}&{ \cdots }&{0} \end{array}} \right] }} \end{array}
矩阵数乘:
\begin{array}{*{20}{l}} {A={ \left[ {\begin{array}{*{20}{c}} {\mathop{{a}}\nolimits_{{11}}}&{\mathop{{a}}\nolimits_{{12}}}&{ \cdots }&{\mathop{{a}}\nolimits_{{1n}}}\\ {\mathop{{a}}\nolimits_{{21}}}&{\mathop{{a}}\nolimits_{{22}}}&{ \cdots }&{\mathop{{a}}\nolimits_{{2n}}}\\ { \vdots }&{ \vdots }&{ \ddots }&{ \vdots }\\ {\mathop{{a}}\nolimits_{{n1}}}&{\mathop{{a}}\nolimits_{{n2}}}&{ \cdots }&{\mathop{{a}}\nolimits_{{nn}}} \end{array}} \right] }}\\ { \alpha A={ \left[ {\begin{array}{*{20}{c}} { \alpha \mathop{{a}}\nolimits_{{11}}}&{ \alpha \mathop{{a}}\nolimits_{{12}}}&{ \cdots }&{ \alpha \mathop{{a}}\nolimits_{{1n}}}\\ { \alpha \mathop{{a}}\nolimits_{{21}}}&{ \alpha \mathop{{a}}\nolimits_{{22}}}&{ \cdots }&{ \alpha \mathop{{a}}\nolimits_{{2n}}}\\ { \vdots }&{ \vdots }&{ \ddots }&{ \vdots }\\ { \alpha \mathop{{a}}\nolimits_{{n1}}}&{ \alpha \mathop{{a}}\nolimits_{{n2}}}&{ \cdots }&{ \alpha \mathop{{a}}\nolimits_{{nn}}} \end{array}} \right] }} \end{array}
线性变化:
{\begin{array}{*{20}{l}} {x \in \mathop{{ \mathbb{R} }}\nolimits^{{n}}}\\ {Ax={ \left[ {\begin{array}{*{20}{c}} {\mathop{{a}}\nolimits_{{11}}}&{\mathop{{a}}\nolimits_{{12}}}&{ \cdots }&{\mathop{{a}}\nolimits_{{1n}}}\\ {\mathop{{a}}\nolimits_{{21}}}&{\mathop{{a}}\nolimits_{{22}}}&{ \cdots }&{\mathop{{a}}\nolimits_{{2n}}}\\ { \vdots }&{ \vdots }&{ \ddots }&{ \vdots }\\ {\mathop{{a}}\nolimits_{{m1}}}&{\mathop{{a}}\nolimits_{{m2}}}&{ \cdots }&{\mathop{{a}}\nolimits_{{mn}}} \end{array}} \right] }{ \left[ {\begin{array}{*{20}{c}} {\mathop{{x}}\nolimits_{{1}}}\\ {\mathop{{x}}\nolimits_{{2}}}\\ { \vdots }\\ {\mathop{{x}}\nolimits_{{n}}} \end{array}} \right] }={ \left[ {\begin{array}{*{20}{c}} {\mathop{{a}}\nolimits_{{11}}\mathop{{x}}\nolimits_{{1}}+\mathop{{a}}\nolimits_{{12}}\mathop{{x}}\nolimits_{{2}}+ \cdots +\mathop{{a}}\nolimits_{{1n}}\mathop{{x}}\nolimits_{{n}}}\\ {\mathop{{a}}\nolimits_{{21}}\mathop{{x}}\nolimits_{{1}}+\mathop{{a}}\nolimits_{{22}}\mathop{{x}}\nolimits_{{2}}+ \cdots +\mathop{{a}}\nolimits_{{2n}}\mathop{{x}}\nolimits_{{n}}}\\ { \vdots }\\ {\mathop{{a}}\nolimits_{{m1}}\mathop{{x}}\nolimits_{{1}}+\mathop{{a}}\nolimits_{{m2}}\mathop{{x}}\nolimits_{{2}}+ \cdots +\mathop{{a}}\nolimits_{{mn}}\mathop{{x}}\nolimits_{{n}}} \end{array}} \right] } \in \mathop{{ \mathbb{R} }}\nolimits^{{n}}}\\ {x,y \in \mathop{{ \mathbb{R} }}\nolimits^{{n}}}\\ {A{ \left( { \alpha x+ \beta y} \right) }= \alpha Ax+ \beta Ay} \end{array}}
线性代数方程组:
\begin{array}{*{20}{l}} { \left\{ {\begin{array}{*{20}{c}} {\mathop{{a}}\nolimits_{{11}}\mathop{{x}}\nolimits_{{1}}+\mathop{{a}}\nolimits_{{12}}\mathop{{x}}\nolimits_{{2}}+ \cdots +\mathop{{a}}\nolimits_{{1n}}\mathop{{x}}\nolimits_{{n}}=\mathop{{b}}\nolimits_{{1}}}\\ {\mathop{{a}}\nolimits_{{21}}\mathop{{x}}\nolimits_{{1}}+\mathop{{a}}\nolimits_{{22}}\mathop{{x}}\nolimits_{{2}}+ \cdots +\mathop{{a}}\nolimits_{{21}}\mathop{{x}}\nolimits_{{n}}=\mathop{{b}}\nolimits_{{2}}}\\ { \vdots }\\ {\mathop{{a}}\nolimits_{{m1}}\mathop{{x}}\nolimits_{{1}}+\mathop{{a}}\nolimits_{{m2}}\mathop{{x}}\nolimits_{{2}}+ \cdots +\mathop{{a}}\nolimits_{{mn}}\mathop{{x}}\nolimits_{{n}}=\mathop{{b}}\nolimits_{{m}}} \end{array}}\right. }\\ {\text{可}\text{写}\text{作}\text{如}\text{下}\text{线}\text{性}\text{变}\text{换}\text{的}\text{形}\text{式}}\\ {x \in \mathop{{ \mathbb{R} }}\nolimits^{{n}}}\\ {Ax={ \left[ {\begin{array}{*{20}{c}} {\mathop{{a}}\nolimits_{{11}}\mathop{{x}}\nolimits_{{1}}+\mathop{{a}}\nolimits_{{12}}\mathop{{x}}\nolimits_{{2}}+ \cdots +\mathop{{a}}\nolimits_{{1n}}\mathop{{x}}\nolimits_{{n}}=\mathop{{b}}\nolimits_{{1}}}\\ {\mathop{{a}}\nolimits_{{21}}\mathop{{x}}\nolimits_{{1}}+\mathop{{a}}\nolimits_{{22}}\mathop{{x}}\nolimits_{{2}}+ \cdots +\mathop{{a}}\nolimits_{{21}}\mathop{{x}}\nolimits_{{n}}=\mathop{{b}}\nolimits_{{2}}}\\ { \vdots }\\ {\mathop{{a}}\nolimits_{{m1}}\mathop{{x}}\nolimits_{{1}}+\mathop{{a}}\nolimits_{{m2}}\mathop{{x}}\nolimits_{{2}}+ \cdots +\mathop{{a}}\nolimits_{{mn}}\mathop{{x}}\nolimits_{{n}}=\mathop{{b}}\nolimits_{{m}}} \end{array}} \right] }={ \left[ {\begin{array}{*{20}{c}} {\mathop{{a}}\nolimits_{{11}}}&{\mathop{{a}}\nolimits_{{12}}}&{ \cdots }&{\mathop{{a}}\nolimits_{{1n}}}\\ {\mathop{{a}}\nolimits_{{21}}}&{\mathop{{a}}\nolimits_{{22}}}&{ \cdots }&{\mathop{{a}}\nolimits_{{21}}}\\ { \vdots }&{ \vdots }&{ \ddots }&{ \vdots }\\ {\mathop{{a}}\nolimits_{{m1}}}&{\mathop{{a}}\nolimits_{{m2}}}&{ \cdots }&{\mathop{{a}}\nolimits_{{mn}}} \end{array}} \right] }{ \left[ {\begin{array}{*{20}{c}} {\mathop{{x}}\nolimits_{{1}}}\\ {\mathop{{x}}\nolimits_{{2}}}\\ { \vdots }\\ {\mathop{{x}}\nolimits_{{n}}} \end{array}} \right] }= \left[ {\begin{array}{*{20}{c}} {\mathop{{b}}\nolimits_{{1}}}\\ {\mathop{{b}}\nolimits_{{2}}}\\ { \vdots }\\ {\mathop{{b}}\nolimits_{{n}}} \end{array}} \right] } \end{array}
运算法则:
\begin{array}{*{20}{l}} {A+B=B+A}\\ { \left( {A+B} \left) +C=A+{ \left( {B+C} \right) }\right. \right. }\\ { \left( { \alpha \beta } \left) A= \alpha { \left( { \beta A} \right) }= \beta { \left( { \alpha A} \right) }\right. \right. }\\ { \alpha { \left( {AB} \right) }={ \left( { \alpha A} \right) }B=A{ \left( { \alpha B} \right) }}\\ { \left( {AB} \left) C=A{ \left( {BC} \right) }=ABC\right. \right. }\\ {\mathop{{ \left( {\mathop{{A}}\nolimits^{{T}}} \right) }}\nolimits^{{T}}=A}\\ {\mathop{{ \left( {A+B} \right) }}\nolimits^{{T}}=\mathop{{A}}\nolimits^{{T}}+\mathop{{B}}\nolimits^{{T}}}\\ {\mathop{{ \left( { \alpha A} \right) }}\nolimits^{{T}}= \alpha \mathop{{A}}\nolimits^{{T}}}\\ {\mathop{{ \left( {AB} \right) }}\nolimits^{{T}}=\mathop{{B}}\nolimits^{{T}}\mathop{{A}}\nolimits^{{T}}}\\ {\mathop{{ \left( {ABC} \right) }}\nolimits^{{T}}=\mathop{{C}}\nolimits^{{T}}\mathop{{B}}\nolimits^{{T}}\mathop{{A}}\nolimits^{{T}}} \end{array}
矩阵转置:
\begin{array}{*{20}{l}} {\mathop{{A}}\nolimits_{{m \times n}}={\mathop{{ \left[ {\begin{array}{*{20}{c}} {\mathop{{a}}\nolimits_{{11}}}&{\mathop{{a}}\nolimits_{{12}}}&{ \cdots }&{\mathop{{a}}\nolimits_{{1n}}}\\ {\mathop{{a}}\nolimits_{{21}}}&{\mathop{{a}}\nolimits_{{22}}}&{ \cdots }&{\mathop{{a}}\nolimits_{{2n}}}\\ { \vdots }&{ \vdots }&{ \ddots }&{ \vdots }\\ {\mathop{{a}}\nolimits_{{m1}}}&{\mathop{{a}}\nolimits_{{m2}}}&{ \cdots }&{\mathop{{a}}\nolimits_{{mn}}} \end{array}} \right] }}\nolimits_{{m \times n}}}}\\ {A \prime =\mathop{{A}}\nolimits^{{T}}=\mathop{{ \left[ {\begin{array}{*{20}{c}} {\mathop{{a}}\nolimits_{{11}}}&{\mathop{{a}}\nolimits_{{21}}}&{ \cdots }&{\mathop{{a}}\nolimits_{{m1}}}\\ {\mathop{{a}}\nolimits_{{12}}}&{\mathop{{a}}\nolimits_{{22}}}&{ \cdots }&{\mathop{{a}}\nolimits_{{m2}}}\\ { \vdots }&{ \vdots }&{ \ddots }&{ \vdots }\\ {\mathop{{a}}\nolimits_{{10}}}&{\mathop{{a}}\nolimits_{{2n}}}&{ \cdots }&{\mathop{{a}}\nolimits_{{mn}}} \end{array}} \right] }}\nolimits_{{n \times m}}} \end{array}
吴文中公式编辑器
大学
01 \ 线性代数
02 \ 概率论
03 \ 无穷级数
04 \ 微积分
05 \ 代数
06 \ 有理函数
07 \ 常微分方程
高中
01 \ 基础知识
02 \ 立体几何
03 \ 数列
04 \ 向量
05 \ 方程
06 \ 导数
07 \ 初等函数
08 \ 不等式
初中
01 \ 基础代数
02 \ 解析几何
03 \ 乘法与因式分解
04 \ 几何图形
05 \ 三角函数
06 \ 三角不等式
07 \ 基础数列
08 \ 一元二次方程
小学
01 \ 单位换算
02 \ 几何图形计算
03 \ 分数
04 \ 数量关系计算
05 \ 算术定律
06 \ 应用题公式
LaTex:其它
01 \ 艺术史
02 \ 音乐理论
03 \ 工作室艺术
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