# Abstract Algebra/Vector Spaces

## DefinitionEdit

Definition (Vector Space)
Let F be a field. A set V with two binary operations: + (addition) and $\times$ (scalar multiplication), is called a Vector Space if it has the following properties:
1. $(V, +)$ forms an abelian group
2. $v(a+b) = va + vb$ for $v \in V$ and $a, b \in F$
3. $a (v + u) = av + au$ for $v,u \in V$ and $a \in F$
4. $(ab)v = a(bv)$
5. $1_F v = v$

The scalar multiplication is formerly defined by $F \times V \xrightarrow{\phi} V$, where $\phi((f,v)) = fv \in V$.

Elements in F are called scalars, while elements in V are called vectors.

Some Properties of Vector Spaces
1. $0_F v = 0_V= a0_V$
2. $(-1_F)v = -v$
3. $av = 0 \iff a = 0 \text{ or } v = 0$
Proofs:
1. $0_F v = 0_F v + 0_V = (0_F + 0_F) v = 0_F v + 0_F v \Rightarrow 0_V = 0_F v . Also, a 0_V = a 0_V + 0_V = a(0_V + 0_V) = a0_V v + a0_V \Rightarrow a0_V = 0_V v$
2. We want to show that $v + (-1_F)v = 0$, but $v + (-1_F)v = 1_F v + (-1_F)v = (1_F + (-1_F))v = 0_Fv = 0_V$
3. Suppose $av = 0$ such that $a \neq 0$, then $a^{-1} (a v) = a^{-1} 0 = 0 \Rightarrow 1_Fv = v = 0$