1._本章公式汇总 - 线性代数

标准正交化

将一线性无关的向量组 (以 $\alpha_1, \alpha_2, \alpha_3$ 为例) 化为标准正交向量组的方法: 步骤 1. 施密特正交化:

\begin{aligned} & \boldsymbol{\beta}_1=\boldsymbol{\alpha}_1, \\ & \boldsymbol{\beta}_2=\boldsymbol{\alpha}_2-\frac{\left(\boldsymbol{\alpha}_2, \boldsymbol{\beta}_1\right)}{\left(\boldsymbol{\beta}_1, \boldsymbol{\beta}_1\right)} \boldsymbol{\beta}_1, \\ & \boldsymbol{\beta}_3=\boldsymbol{\alpha}_3-\frac{\left(\boldsymbol{\alpha}_3, \boldsymbol{\beta}_1\right)}{\left(\boldsymbol{\beta}_1, \boldsymbol{\beta}_1\right)} \boldsymbol{\beta}_1-\frac{\left(\boldsymbol{\alpha}_3, \boldsymbol{\beta}_2\right)}{\left(\boldsymbol{\beta}_2, \boldsymbol{\beta}_2\right)} \boldsymbol{\beta}_2 . \end{aligned}

步骤 2. 规范化（单位化):

\gamma_1=\frac{\boldsymbol{\beta}_1}{\left\|\boldsymbol{\beta}_1\right\|}, \gamma_2=\frac{\boldsymbol{\beta}_2}{\left\|\boldsymbol{\beta}_2\right\|}, \boldsymbol{\gamma}_3=\frac{\boldsymbol{\beta}_3}{\left\|\boldsymbol{\beta}_3\right\|} .