---
# System prepended metadata

title: Vector Space
tags: [vector space, math]

---

{%hackmd @RintarouTW/About %}

# Vector Space


### + : $V+V\to V$

#### Associative
$$
\forall x,y,z \in V, (x+y)+z = x+(y+z)
$$

#### Communative
$$
\forall x,y\in V, x+y=y+x
$$

#### Inverse
$$
\forall x\in V, \exists -x, -x + x = 0
$$

#### Identity
$$
\forall x\in V, x+0 = x
$$

### Scaler Multiplication

$a,b\in \mathbb{R}, x,y\in V$

$$
a(x+y) = ax+ay
$$

$$
(a+b)x = ax+bx
$$

$$
a(bx) = (ab)x
$$

$$
1x = x
$$

## Linear Combination

$\forall y\in V,\exists a_1,a_2,\cdots,a_n\in \mathbb{R}, x_1,x_2,\cdots,x_n\in V$

$$
y = a_1x_1+a_2x_2+\cdots+a_nx_n
$$

## Basis

### Linear Indpendent

$$
a_1x_1+a_2x_2+\cdots+a_nx_n=0\implies a_1=a_2=\cdots=a_n=0
$$

### Basis Size is Constant

$V$ is a vector space with a basis containing $n$ elements, then all bases of $V$ contain $n$ elements.

### Dimention of $V$

$V$ over $\mathbb{R}$ with basis {$x_1,x_2,\cdots,x_n$}, dim $V$ = $n$


## Group Isomorphic

- bijection
- $f(x\cdot_{A} y) = f(x)\cdot_{B} f(y)$

###### tags: `math` `vector space`