1. Permutations P:
execute row exchanges
becomes PA = LU for any invertible A
Permutations P = identity matrix with reordered rows
m=n (n-1) ... (3) (2) (1) counts recordings, counts all nxn permuations
对于nxn矩阵存在着n!个置换矩阵
, 
2. Transpose:
2.1 Symmetric matrices
对称矩阵 
2.2 矩阵乘积的转置
2.3 
is always symmetric
why? take transpose 
3. 向量空间 Vector spaces
向量空间对线性运算封闭，即空间内向量进行线性运算得到的向量仍在空间之内
example: 
= all 2-dim real vectors=x-y plane
first component, second component
= all vectors with 3 components
= all column vectors with m real components
所有向量空间必然包含零向量，因为任何向量数乘0或者加上反向量都会得到零向量，而因为向量空间对线性运算封闭，所以零向量必属于向量空间
反例 not a vector space:
中的第一象限则不是一个向量空间, 加法数乘不封闭
4. 子空间 Subspaces
a vector space inside , subspace of
line in through zero vector
反例:
中不穿过原点的直线就不是向量空间。子空间必须包含零向量，原因就是数乘0的到的零向量必须处于子空间中
subspaces of :
-
all of
-
any line through
L(line) -
zero vector only z(zero)
subspaces of :
-
all of
-
any plane through
P(plane) -
any line through
L(line) -
zero vector only z(zero) =
5. 列空间 Column spaces
Columns in : all their combinations from a subspace called column space C(A)
空间内包含两向量的所有线性组合
