學海無涯，思境無維；數乃六藝，理之始也。
或有一得足矣 愚千慮

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Vector Space

+ :

$V+V\to V$

Associative

$$\mathrm{\forall }x,y,z\in V,(x+y)+z=x+(y+z)$$

Communative

$$\mathrm{\forall }x,y\in V,x+y=y+x$$

Inverse

$$\mathrm{\forall }x\in V,\mathrm{\exists }-x,-x+x=0$$

Identity

$$\mathrm{\forall }x\in V,x+0=x$$

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

$\mathrm{\forall }y\in V,\mathrm{\exists }{a}_1,a_2,\cdots ,a_n\in \mathbb{R},x_1,x_2,\cdots ,x_n\in V$

$$y=a_1{x}_1+a_2{x}_2+\cdots +a_n{x}_n$$

Basis

Linear Indpendent

$$a_1{x}_1+a_2{x}_2+\cdots +a_n{x}_n=0⟹a_1=a_2=\cdots =a_n=0$$

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