Vector Space

A vector space $V$ over a Field is a set of vectors that has:

• Closed Under Addition
• Closed Under Scalar Multiplication And both operations follows the 8 axioms:

1. $⊞$ is Associative ($\forall x,y,z\in V,(x+y)+z=x+(y+z)$)
2. $⊞$ is Commutative ($\forall x,y\in V,x+y=y+x$)
3. Additive Identity ($\exists \overrightarrow{0}\in V$ s.t $\forall x\in V,\overrightarrow{0}+\overrightarrow{x}=\overrightarrow{x}$)
4. Additive Inverse for all $x$. ($\forall x\in V,\exists x^{\prime }$ s.t $x+x^{\prime }=0$)
5. Associative scalar multiplication $\forall x\in V,\forall c,d\in F,(cd)\overrightarrow{x}=c(d\overrightarrow{x})$
6. Distributive under addition of vectors ($c⊡(\overrightarrow{x}⊞\overrightarrow{y})$ = $(c⊡\overrightarrow{x})⊞(c⊡\overrightarrow{y})$)
7. Distributive under addition of scalars ($\forall x\in V,\forall c,d\in F,(c+d)(\overrightarrow{x})=c\overrightarrow{x}+d\overrightarrow{x}$)
8. Multiplicative Identity $\forall x\in V,1\overrightarrow{x}=\overrightarrow{x}$

Checking Valid Vector Spaces

You need to check that all 8 axioms apply to the vector space.

Propositions

1. The zero vector $0$ is unique
2. $\forall x\in V,0x=0$
3. $\forall x\in V,x^{\prime }$ is unique
4. $\forall x\in V,\forall c\in R,(-c)x=-(cx)$

Concepts

• Subspace
• Vector Space Dimension
• Finite Dimension Vector Space Implications