• 内积
• 投影
• 向量的线性相关/线性无关
• 向量空间的基
• 线性变换和线性映射
• 矩阵降维
• 特征值和特征向量
• 正交向量组和正交矩阵
• 实对称矩阵

内积

代数运算

$[\alpha, \beta] = a_{1}b_{1} + a_{2}b_{2} + ... + a_{n}b_{n}$

几何运算

$||\alpha|| = \sqrt{[\alpha, \alpha]} = \sqrt{a_{1}^{2} + a_{2}^{2} + ... + a_{n}^{2}}$

$$\alpha, \beta$$$$n$$元实非零向量，记

$< \alpha, \beta > = \arccos \frac {[\alpha, \beta]}{||\alpha|| ||\beta||}, 0\leq <\alpha, \beta> \leq \pi$

$$<\alpha, \beta>$$为向量$$\alpha$$$$\beta$$的夹角

$\alpha\cdot \beta = ||\alpha|| ||\beta|| \cos \theta$

投影

投影和内积关系

$\alpha \cdot \beta = ||\alpha||\cdot \cos (\theta)$

向量的线性相关/线性无关

$k_{1}\alpha_{1} + k_{2}\alpha_{2} + ... + k_{m}\alpha_{m} = 0$

向量空间的基

$$V$$是一个向量空间，$$\alpha_{1},\alpha_{2},...,\alpha_{r}$$$$V$$中的一组向量，如果满足

1. $$\alpha_{1},\alpha_{2},...,\alpha_{r}$$线性无关
2. $$V$$中的任一向量都可由$$\alpha_{1},\alpha_{2},...,\alpha_{r}$$线性表示

基变换与坐标变换

$$\alpha_{1}, \alpha_{2}, ..., \alpha_{r}$$$$\beta_{1}, \beta_{2}, ..., \beta_{r}$$为向量空间$$V$$的基，有

$(\beta_{1}, \beta_{2}, ..., \beta_{r}) = (\alpha_{1}, \alpha_{2}, ..., \alpha_{r}) P_{r\times r}$

$$r$$阶矩阵$$P$$是由基$$\alpha_{1}, \alpha_{2}, ..., \alpha_{r}$$到基$$\beta_{1}, \beta_{2}, ..., \beta_{r}$$的过渡矩阵，称上式为基变换公式

$$V$$是向量空间，$$\alpha_{1}, \alpha_{2}, ..., \alpha_{r}$$$$\beta_{1}, \beta_{2}, ..., \beta_{r}$$分别为$$V$$的基，且

$X = (x_{1}, x_{2}, ..., x_{r})^{T}\\ Y = (y_{1}, y_{2}, ..., y_{r})^{T}$

$X = PY \ 或\ Y = P^{-1}X$

矩阵降维

$PA= \begin{pmatrix} p_{1}\\ p_{2}\\ \vdots \\ p_{M} \end{pmatrix} (a_{1}, a_{2}, \cdots, a_{R}) \in R^{M\times R}$

特征值和特征向量

$$A$$$$n$$阶方阵，若存在数$$\lambda$$$$n$$维非零向量$$X$$，使得

$AX = \lambda X$

性质

$\sum_{i=1}^{n} \lambda_{i} =\sum_{i=1}^{n} a_{ii}$

$\prod_{i=1}^{n} \lambda_{i} =|A|$

求解

1. 计算$$A$$的特征多项式：$$f(\lambda) = | A - \lambda E |$$，其根$$\lambda_{1}, \lambda_{2}, ..., \lambda_{s}(\lambda_{i} \neq \lambda_{j})$$就是$$A$$$$s$$个不同的特征值

2. 对每个特征值$$\lambda_{i}, i=1,2,...,s$$，解方程组$$(A-\lambda_{i}E)X=0$$，其基础解系就是$$A$$的对应于特征值$$\lambda_{i}$$的线性无关的特征向量，其非零解就是$$A$$的对应于特征值$$\lambda_{i}$$的全部特征向量

正交向量组和正交矩阵

$$\alpha, \beta$$是两个$$n$$元实向量，若$$[\lambda, \beta]=0$$，则称$$\lambda$$$$\beta$$正交（或垂直），记为

$$A$$$$n$$阶矩阵，如果$$AA^{T}=E$$，则称$$A$$为正交矩阵

$$A$$为正交矩阵的充分必要条件是$$A$$的列（或行）向量组是单位正交向量组

实对称矩阵

$Q^{-1}AQ = Q^{T}AQ = \Lambda = diag(\lambda_{1}, \lambda_{2}, ..., \lambda_{n})$