05 MIT线性代数-转置,置换,向量空间Transposes, permutations, spaces

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 :

1. all of

2. any line through L(line)

3. zero vector only z(zero)

subspaces of :

1. all of

2. any plane through P(plane)

3. any line through L(line)

4. 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