Linear space (Algebra)

1.Definition
{V,P,+,*} satisfies the eight rules

2.dimension, base and coordinate

3.change of the base

relationship:
e1,e2, ..., en and e'1, e'2, ...,e'n are two group of bases,
e'1=a11*e1+a21*e2+...+an1*en
e'2=a12*e1+a22*e2+...+an2*en
....
e'n=a1n*e1+a2n*e2+...+ann*en

vector e=x1*e1+x2*e2+...+xn*en=x'1*e'1+x'2*e'2+...+x'n*e'n
=(e1,e2, ..., en )*X=(e'1, e'2, ...,e'n )*X'

then A=[a11, a12, ..., a1n; ...; an1, an2, ..., ann] is transition matrix (full rank)
coordinate: X=AX'
bases: (e'1, e'2, ...,e'n )=(e1,e2, ..., en )*A

4. linear subspace
key points: non-empty, closure
Th. equivalent vectors span the same subspace; them have the same rank
Th. Any m-dimensional subspace can extends to the whole n-dimensional space

5.Joint and Sum
definition of joint and sum
dimensional formula: D(V1)+D(V2)=D(V1+V2)+D(V1&V2) (proof)

6.Direct Sum
Every vector has Unique decomposition: a=a1+a2 where a1belongs to V1 and a2 belongs to V2
<==>Zero vector has unique decomposition
<==>V1&V2={0}
<==>if W=V1+V2, then D(W)=D(V1)+D(V2)

Always exists W, for U to make V=U Direct Sum W
The conception can be extended to multiple spaces.

8 同构**
向量与其坐标的对应实质是V 到 Pn的1-1对应。
同构：V到V'有一个1-1的映上的映射满足向量的可加性和数乘向量的均匀性。、
同构映射的逆和同构映射的乘积也是同构
同构是一种等价关系（反身，对称，传递）
P上任意n维线性空间都与Pn同构；P上任意两个n维线性空间都同构；P上两个有限维线性空间同构的充要条件是有相同维数
总之，同构的线性空间可以不加区别，维数是有限维线性空间的唯一本质特征。