Section01_定义

1. 向量，零向量，模
   1. $\boldsymbol{\alpha} = \left( \begin{matrix} a_{1} \\ \vdots \\ a_{n} \end{matrix} \right)$ 为 $n$ 维列向量，
   2. $\boldsymbol{\alpha} = \left( \begin{matrix} 0 \\ \vdots\\ 0 \end{matrix} \right)$ 为零向量
   3. $\vert \boldsymbol{\alpha} \vert = \sqrt{a_{1}^{2}+a_{2}^{2}+\cdots+a_{n}^{2}} = \sqrt{\boldsymbol{\alpha}^{\intercal}\boldsymbol{\alpha}}$ 称为"大小"，"长度"，"模"
      1. $\vert \boldsymbol{\alpha} \vert = 0$，则 $\boldsymbol{\alpha} = \boldsymbol{0}$
      2. $\vert \boldsymbol{\alpha} \vert = 1$，则 $\boldsymbol{\alpha}$ 为单位向量(规范向量)
      3. $\boldsymbol{\alpha} \ne \boldsymbol{0},\ \boldsymbol{\alpha}^{\circ} = \frac{1}{\vert \boldsymbol{\alpha} \vert}\cdot \boldsymbol{\alpha}$ 称为单位化
2. 内积 对于 $\boldsymbol{\alpha} = \left( \begin{matrix} a_{1} \\ \vdots \\ a_{n} \end{matrix} \right), \boldsymbol{\beta} = \left( \begin{matrix} b_{1} \\ \vdots \\ b_{n} \end{matrix} \right)$，$(\boldsymbol{\alpha}, \boldsymbol{\beta}) = a_{1}b_{1} + a_{2}b_{2} + \cdots + a_{n}b_{n}$ 称为向量的内积

   Notes

   1. $(\boldsymbol{\alpha}, \boldsymbol{\beta}) = (\boldsymbol{\beta}, \boldsymbol{\alpha}) = \boldsymbol{\alpha}^{\intercal}\boldsymbol{\beta} = \boldsymbol{\beta}^{\intercal}\boldsymbol{\alpha}$
   2. $(\boldsymbol{\alpha}, \boldsymbol{\alpha}) = \boldsymbol{\alpha}^{\intercal}\boldsymbol{\alpha} = \vert \boldsymbol{\alpha} \vert^{2}$
   3. $(\boldsymbol{\alpha}, k_{1}\boldsymbol{\beta}_{1} + k_{2}\boldsymbol{\beta}_{2} + \cdots + k_{s}\boldsymbol{\beta}_{s})= \sum\limits_{i=1}^{s}k_{i}(\boldsymbol{\alpha}, \boldsymbol{\beta}_{i})$
   4. $(\boldsymbol{\alpha},\boldsymbol{\beta}) = 0\Leftrightarrow \boldsymbol{\alpha}\perp \boldsymbol{\beta}$，若 $(\boldsymbol{\alpha},\boldsymbol{\beta}) = 0$，称 $\boldsymbol{\alpha}$ 与 $\boldsymbol{\beta}$ 正交，记为 $\boldsymbol{\alpha}\perp \boldsymbol{\beta}$

   5. 零向量同任何向量正交，也同任何向量平行