Abstract Algebra/Vector Spaces


 * 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:

The scalar multiplication is formally defined by $$F \times V \xrightarrow{\phi} V$$, where $$\phi((f,v)) = fv \in V$$.
 * 1) $$(V, +)$$ forms an abelian group
 * 2) $$(a + b)v = av + bv$$ for $$a, b \in F$$ and $$v \in V$$
 * 3) $$a(v + u) = av + au$$ for $$a \in F$$ and $$v,u \in V$$
 * 4) $$(ab)v = a(bv)$$
 * 5) $$1_F v = v$$

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) $$0_F v = (0_F + 0_F) v = 0_F v + 0_F v \Rightarrow 0_V = 0_F v . Also, a 0_V = a(0_V + 0_V) = a0_V + a0_V \Rightarrow a0_V = 0_V$$
 * 2) We want to show that $$v + (-1_F)v = 0_V$$, 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 $$a^{-1} (a v) = a^{-1} 0 = 0 \Rightarrow 1_Fv = v = 0$$