# Schedule

• PA=LU
• Vector Spaces and Subspaces

# Permutations – 置换矩阵

Permutations P : execute row exchanges.

Permutations P is the identity matrix with reordered rows.

# Transpose – 转置矩阵

$A^T$中第$i$行第$j$列的元素等于$A$中第$j$行第$i$列的元素,即$(A^T){ij}=A{ji}$.

# Symmetric Matrix – 对称矩阵

e.g.
\begin{bmatrix}1&3\2&3\4&1\end{bmatrix}*\begin{bmatrix}1&2&4\3&3&1\end{bmatrix}=\begin{bmatrix}10&11&7\11&13&11\7&11&17\end{bmatrix}

(R^TR)^T=R^T(R^T)^T=R^TR

# Vector Spaces – 向量空间

e.g.
R^2=\text{all 2-dim real vectors} = \text{x-y plane}, 例如: \begin{bmatrix}3\2\end{bmatrix},\begin{bmatrix}0\0\end{bmatrix},\begin{bmatrix}\pi\e\end{bmatrix}
R^3= \text{all vectors with 3 components}, 例如:\begin{bmatrix}3\2\0\end{bmatrix}
R^n= \text{all column with n(n is real number) components}

A vector space has to be closed under multiplication and addition of vectors. In other words, linear combinations.

1. all vectors of R^2 所有的二维向量
2. any line through \begin{bmatrix}0\0\end{bmatrix} 任意过原点的直线
3. zero vector(零向量), Z=\begin{bmatrix}0\0\end{bmatrix} 仅含零向量的空间
同理,R^3的子空间有:
4. all vectors of R^3 所有的三维空间
5. any plane through \begin{bmatrix}0\0\0\end{bmatrix} 任意过原点的平面
6. any line through \begin{bmatrix}0\0\0\end{bmatrix} 任意过原点的直线
7. zero vector(零向量), Z=\begin{bmatrix}0\0\0\end{bmatrix} 仅含零向量的空间

# Subspace – 如何构造子空间?

A=\begin{bmatrix}1&3\2&3\4&1\end{bmatrix}

All these linear combinations form a subspace.