线性代数

未完待续

Vector Space

$V$: a set(vector)
$F$: field(scalan)
$\times,+$

• (VS-1) 加法封闭性
• (VS 0) 乘法封闭性
• (VS 1) 加法有交换律
• (VS 2) 加法有结合律
• (VS 3) 存在加法单位元素
• (VS 4) 存在加法反元素
• (VS 5) 存在乘法单位元素
• (VS 6,7,8) 分配律

Def
Let $V$ be a vector space over a field $F$ and $W\subset V$.Then $W$ is a subspace of $V$ if $W$ is a vector space over $F$ with the same two operations.
Def
两个集合的加法

$$S_1+S_2=\{x+y;\ x\in S_1,\ y\in S_2\}$$

Polynomial

$$P(F)={a_n\boldsymbol{x}^n+...+a_1\boldsymbol{x}+a_0; n\in N,a_i\in F,}\\ p_n(F)=$$

span

$$Span(S)=\{\sum_{i=1}^{n}a_iu_i; a_i\in F,u_i\in S,i=1,2,...,n\}$$

$
let\ W_1,W_2 \ be \ subspaces \ of \ V \\
\Rightarrow\\
1)\ W_1+W_2,\ is\ a\ subspace\ of\ V.\\
\ \ \ \ \ 2)\ W_1\cap W_2,\ \ is\ a\ subspace\ of\ V.\\
\ \ \ \ \ 3)\ W_1\cup W_2,\ \ is\ not\ necessary\ a\ subspace\ of\ V.\\
\ \ \ \ \ 4)\ W_1+ W_2,\ \ is\ the\ “smallest”\ subspace\ containing\ W_1\ and\ W_2.\\
\ \ \ \ \ 2)\ W_1\cup W_2,\ \ is\ a\ subspace\ of\ V.\\
\ \ \ \ \ \ \ \ \ \ \Leftrightarrow W_1\subset W_2\ or\ W_2\subset W_1.
$

trivial 简单 properties 性质 Span(s) Polynomial多项式