抽象代数入门（四）

向量空间(vector space)

Definition of vector space: a vector space is a set closed under vector additon and scalar multiplication.

1. 满足加法交换律、结合律、分配律，存在加法单位元
2. 满足标量乘法的交换律、结合律、分配律，存在乘法单位元

Cyclic Subspace(循环子空间)

Dfn: 循环子空间是通过有限维的向量空间和一个线性变换构成的。

 

A cyclic subspace of  can be defined as:

, 

.  is A-invariant subspace of .

prf: , V is A-invariant, . Au=A(pv)

Rem:

1. A 的特征向量同以为的A-invariant 子空间一一对应。
2. If v is an eigenvector of A,  = Span(v).

Definition: . we all know that every ideal of  is principal. i.e. it can be generated by one element.

,  is the monoic, smallest degree s.t. . () 表示这个多项式生成的向量空间。

注意:

A-invariant

Definition:   is a subspace of .We say "S" is A-invariant if . i.e. .

简写成 , .

Definition: If v is eigenvector of A, 

.