# 向量空间

（重定向自线性空间

${\displaystyle \mathbf {A} ={\begin{bmatrix}1&2\\3&4\end{bmatrix}}}$

## 正式定義

• 向量加法 ${\displaystyle \oplus :V\times V\to V}$  （其中 ${\displaystyle \oplus (u,\,v)}$  慣例上簡記為 ${\displaystyle u\oplus v}$
• 标量乘法 ${\displaystyle \cdot :K\times V\to V}$  （其中 ${\displaystyle \cdot \,(a,\,v)}$  慣例上簡記為 ${\displaystyle a\cdot v}$  甚至是 ${\displaystyle av}$

## 基本性质

${\displaystyle a\cdot 0_{V}=a\cdot (0_{V}+0_{V})=a\cdot 0_{V}+a\cdot 0_{V}}$

{\displaystyle {\begin{aligned}0_{V}&=u+a\cdot 0_{V}\\&=u+(a\cdot 0_{V}+a\cdot 0_{V})\\&=(u+a\cdot 0_{V})+a\cdot 0_{V}\\&=0_{V}+a\cdot 0_{V}\\&=a\cdot 0_{V}\\\end{aligned}}}

${\displaystyle 0_{K}\cdot u=(0_{K}+0_{K})\cdot u=0_{K}\cdot u+0_{K}\cdot u}$

${\displaystyle 0_{V}=w+0_{K}\cdot u=w+(0_{K}\cdot u+0_{K}\cdot u)=(w+0_{K}\cdot u)+0_{K}\cdot u=0_{V}+0_{K}\cdot u=0_{K}\cdot u}$

${\displaystyle K=\{0_{K}\}}$  ，根據定理(3)本定理顯然成立。下面只考慮 ${\displaystyle K\neq \{0_{K}\}}$  的狀況。

{\displaystyle {\begin{aligned}u&=1_{K}\cdot u\\&=(a\times {\frac {1}{a}})\cdot u\end{aligned}}}

{\displaystyle {\begin{aligned}u&=(a\times {\frac {1}{a}})\cdot u\\&=({\frac {1}{a}}\times a)\cdot u\\&={\frac {1}{a}}\cdot (a\cdot u)\end{aligned}}}

{\displaystyle {\begin{aligned}u&={\frac {1}{a}}\cdot (a\cdot u)\\&={\frac {1}{a}}\cdot 0_{V}\\&=0_{V}\end{aligned}}}

${\displaystyle 0_{K}\cdot u=(-a+a)\cdot u=-a\cdot u+a\cdot u}$
${\displaystyle 0_{K}\cdot u=[a+(-a)]\cdot u=a\cdot u+(-a)\cdot u}$

${\displaystyle 0_{V}=0_{K}\cdot u=-a\cdot u+a\cdot u}$
${\displaystyle 0_{V}=0_{K}\cdot u=a\cdot u+(-a)\cdot u}$

## 例子

1. u + (v + w) = (u + v) + w
2. v + w = w + v
3. 零元素存在：零元素0满足：对任何的向量元素vv + 0 = v
4. 逆元素存在：对任何的向量元素v，它的相反数w = −v就满足v + w = 0
5. 标量乘法对向量加法满足分配律a(v + w) = a v + a w.
6. 向量乘法对标量加法满足分配律(a + b)v = a v + b v.
7. 标量乘法与标量的域乘法相容：a(bv) =(ab)v
8. 标量乘法有單位元中的乘法单位元，也就是实数“1”满足：对任意实数v1v = v

${\displaystyle \forall \lambda \in \mathbb {R} ,\,v=(a_{1},a_{2},\cdots ,a_{n})\in \mathbb {R} ^{n},\,w=(b_{1},b_{2},\cdots ,b_{n})\in \mathbb {R} ^{n}}$
${\displaystyle v+w=(a_{1},a_{2},\cdots ,a_{n})+(b_{1},b_{2},\cdots ,b_{n})=(a_{1}+b_{1},a_{2}+b_{2},\cdots ,a_{n}+b_{n})}$
${\displaystyle \lambda v=\lambda (a_{1},a_{2},\cdots ,a_{n})=(\lambda a_{1},\lambda a_{2},\cdots ,\lambda a_{n})}$

### 方程组与向量空间

${\displaystyle 3x+2y-z=0}$
${\displaystyle x+5y+2z=0}$

${\displaystyle 3(x_{1}+x_{2})+2(y_{1}+y_{2})-(z_{1}+z_{2})=(3x_{1}+2y_{1}-z_{1})+(3x_{2}+2y_{2}-z_{2})=0}$
${\displaystyle (x_{1}+x_{2})+5(y_{1}+y_{2})+2(z_{1}+z_{2})=(x_{1}+5y_{1}+2z_{1})+(x_{2}+5y_{2}+2z_{2})=0}$

${\displaystyle f''+4xf'+\cos(x)f=0}$

## 子空間基底

${\displaystyle v=\lambda _{1}e_{1}+\lambda _{2}e_{2}+\cdots +\lambda _{n}e_{n}+\cdots }$

${\displaystyle v=\lambda _{1}e_{1}+\lambda _{2}e_{2}+\cdots +\lambda _{n}e_{n}}$

${\displaystyle e_{1}=(1,0,\cdots ,0)}$
${\displaystyle e_{2}=(0,1,\cdots ,0)}$
${\displaystyle e_{n}=(0,0,\cdots ,1)}$

## 線性映射

${\displaystyle f:\,V\rightarrow W}$
${\displaystyle \forall a\in F,u,v\in V,\,f(u+v)=f(u)+f(v),\,f(a\cdot v)=a\cdot f(v)}$

${\displaystyle g\circ f(x)=x,\,f\circ g(y)=y}$

## 參考文獻

• 中国大百科全书
• Howard Anton and Chris Rorres. Elementary Linear Algebra, Wiley, 9th edition, ISBN 0-471-66959-8.
• Kenneth Hoffmann and Ray Kunze. Linear Algebra, Prentice Hall, ISBN 0-13-536797-2.
• Seymour Lipschutz and Marc Lipson. Schaum's Outline of Linear Algebra, McGraw-Hill, 3rd edition, ISBN 0-07-136200-2.
• Gregory H. Moore. The axiomatization of linear algebra: 1875-1940, Historia Mathematica 22 (1995), no. 3, 262-303.
• Gilbert Strang. "Introduction to Linear Algebra, Third Edition", Wellesley-Cambridge Press, ISBN 0-9614088-9-8

## 參考資料

1. ^ Roman 2005, ch. 1, p. 27