3.1 向量的正交

• $\boldsymbol{\alpha}=\begin{bmatrix} a_1,a_2,\cdots,a_n \end{bmatrix}^T,\boldsymbol{\beta}=\begin{bmatrix} b_1,b_2,\cdots,b_n \end{bmatrix}^T$

---

• 內積：$(\boldsymbol{\alpha},\boldsymbol{\beta}) = \boldsymbol{\alpha}^T\boldsymbol{\beta} = \sum\limits_{i=1}^{n}a_ib_i = a_1b_1+a_2b_2+\cdots+a_nb_n$
• 模：$\begin{Vmatrix} \boldsymbol{\alpha} \end{Vmatrix} = \sqrt{\boldsymbol{\alpha}^T\boldsymbol{\alpha}} = \sqrt{\sum\limits_{i=1}^{n}a_i^2} = \sqrt{a_1^2+a_2^2+\cdots+a_n^2}$

---

• 正交：$(\boldsymbol{\alpha},\boldsymbol{\beta})=\boldsymbol{\alpha}^T\boldsymbol{\beta}=0$
• 單位向量：$\begin{Vmatrix} \boldsymbol{\alpha} \end{Vmatrix} = \sqrt{\boldsymbol{\alpha}^T\boldsymbol{\alpha}} = 1$

---

• 標準正交向量組：若列向量 $\boldsymbol{\alpha}_1,\boldsymbol{\alpha}_2,\cdots,\boldsymbol{\alpha}_n$ 滿足 $\boldsymbol{\alpha}_i^T\boldsymbol{\alpha}_j=\begin{cases} 0,&i \ne j\\1,&i=j \end{cases}$（均爲單位向量且任意兩不同向量正交），則稱 $\boldsymbol{\alpha}_1,\boldsymbol{\alpha}_2,\cdots,\boldsymbol{\alpha}_n$ 是標準（單位）正交向量組。
• 線性無關向量組 $\boldsymbol{\alpha}_1,\boldsymbol{\alpha}_2,\cdots,\boldsymbol{\alpha}_n \rightarrow$ 標準正交向量組 $\boldsymbol{\eta}_1,\boldsymbol{\eta}_2,\cdots,\boldsymbol{\eta}_n$：

  1. 正交化得正交向量組 $\boldsymbol{\beta}_1,\boldsymbol{\beta}_2,\cdots,\boldsymbol{\beta}_n$：
     $\begin{aligned} & \boldsymbol{\beta}_1 = \boldsymbol{\alpha}_1,\\ & \boldsymbol{\beta}_2 = \boldsymbol{\alpha}_2 - \frac{(\boldsymbol{\alpha}_2,\boldsymbol{\beta}_1)}{(\boldsymbol{\beta}_1,\boldsymbol{\beta}_1)}\boldsymbol{\beta}_1,\\ & \cdots\\ & \boldsymbol{\beta}_n = \boldsymbol{\alpha}_n - \frac{(\boldsymbol{\alpha}_n,\boldsymbol{\beta}_{n-1})}{(\boldsymbol{\beta}_{n-1},\boldsymbol{\beta}_{n-1})}\boldsymbol{\beta}_{n-1} - \frac{(\boldsymbol{\alpha}_n,\boldsymbol{\beta}_{n-2})}{(\boldsymbol{\beta}_{n-2},\boldsymbol{\beta}_{n-2})}\boldsymbol{\beta}_{n-2} - \cdots - \frac{(\boldsymbol{\alpha}_n,\boldsymbol{\beta}_{1})}{(\boldsymbol{\beta}_{1},\boldsymbol{\beta}_{1})}\boldsymbol{\beta}_{1} \end{aligned}$

  2. 單位化得標準正交向量組 $\boldsymbol{\eta}_1,\boldsymbol{\eta}_2,\cdots,\boldsymbol{\eta}_n$：
     $\begin{aligned} &\boldsymbol{\eta}_1 = \frac{\boldsymbol{\beta}_1}{\begin{Vmatrix} \boldsymbol{\beta}_1 \end{Vmatrix}},\\ &\boldsymbol{\eta}_2 = \frac{\boldsymbol{\beta}_2}{\begin{Vmatrix} \boldsymbol{\beta}_2 \end{Vmatrix}},\\ &\cdots\\ &\boldsymbol{\eta}_n = \frac{\boldsymbol{\beta}_n}{\begin{Vmatrix} \boldsymbol{\beta}_n \end{Vmatrix}} \end{aligned}$