Chapter 45: determinant
https://www.bilibili.com/video/BV13e411m7Js
https://www.youtube.com/watch?v=Sv7VseMsOQc
45.2 determinant axioms
determinant axioms
\[ \begin{aligned} & \begin{cases} \begin{cases} \det\left(\boldsymbol{u},\boldsymbol{v}\right)=\det\left(\boldsymbol{u}+s\boldsymbol{v},\boldsymbol{v}\right)\\ \det\left(\boldsymbol{u},\boldsymbol{v}\right)=\det\left(\boldsymbol{u},\boldsymbol{v}+s\boldsymbol{u}\right) \end{cases} & \text{translation invariance}\\ \begin{cases} \det\left(s\boldsymbol{u},\boldsymbol{v}\right)=s\det\left(\boldsymbol{u},\boldsymbol{v}\right)\\ \det\left(\boldsymbol{u},s\boldsymbol{v}\right)=s\det\left(\boldsymbol{u},\boldsymbol{v}\right) \end{cases} & \text{scaling}\\ \begin{cases} \det\left(\boldsymbol{u}_{{\scriptscriptstyle 1}}+\boldsymbol{u}_{{\scriptscriptstyle 2}},\boldsymbol{v}\right)=\det\left(\boldsymbol{u}_{{\scriptscriptstyle 1}},\boldsymbol{v}\right)+\det\left(\boldsymbol{u}_{{\scriptscriptstyle 2}},\boldsymbol{v}\right)\\ \det\left(\boldsymbol{u},\boldsymbol{v}_{{\scriptscriptstyle 1}}+\boldsymbol{v}_{{\scriptscriptstyle 2}}\right)=\det\left(\boldsymbol{u},\boldsymbol{v}_{{\scriptscriptstyle 1}}\right)+\det\left(\boldsymbol{u},\boldsymbol{v}_{{\scriptscriptstyle 2}}\right) \end{cases} & \text{decomposition} \end{cases}\\ \Leftrightarrow & \begin{cases} \begin{cases} \det\left(\boldsymbol{u},\boldsymbol{v}\right)=\det\left(\boldsymbol{u}+s\boldsymbol{v},\boldsymbol{v}\right)\\ \det\left(\boldsymbol{u},\boldsymbol{v}\right)=\det\left(\boldsymbol{u},\boldsymbol{v}+s\boldsymbol{u}\right) \end{cases} & \text{translation invariance}\\ \begin{cases} \det\left(\boldsymbol{u}_{{\scriptscriptstyle 1}}+s\boldsymbol{u}_{{\scriptscriptstyle 2}},\boldsymbol{v}\right)=\det\left(\boldsymbol{u}_{{\scriptscriptstyle 1}},\boldsymbol{v}\right)+s\det\left(\boldsymbol{u}_{{\scriptscriptstyle 2}},\boldsymbol{v}\right)\\ \det\left(\boldsymbol{u},\boldsymbol{v}_{{\scriptscriptstyle 1}}+s\boldsymbol{v}_{{\scriptscriptstyle 2}}\right)=\det\left(\boldsymbol{u},\boldsymbol{v}_{{\scriptscriptstyle 1}}\right)+s\det\left(\boldsymbol{u},\boldsymbol{v}_{{\scriptscriptstyle 2}}\right) \end{cases} & \text{linearity} \end{cases} \end{aligned} \]