Vector Spaces

A non-empty set $V$ of objects is a vector space if it satisfies the following 10 axioms.

• 1. Closure Axioms [A1 - A2]
• 2. Vector Addition Properties [A3 - A6]
• 3. Scalar Multiplication Properties [A7 - A10]

Subspaces

A subset $U$ of a vector space $V$ is a subspace if it satisfies the following 3 properties.

1. A zero element exists. $\exists O \in U \space | \space \forall x \in U, O + x = x$

2. It is closed under vector addition. $\forall x, y \in U, \exists ! \space z \in U \space | \space x + y = z$

3. It is closed under scalar multiplication. $\forall x \in U, \forall \alpha \in F, \alpha x \in U$

                       An alternative to the first property is that $U$ is non-empty.
   

Dimension

The dimension is the number of parameters needed to describe an element $\mathbf{v}$ of a vector space $V$; the number of vectors in any given basis for $V$.

$$ \mathbf{v} \in V= \R^3 , \hspace{5pt} \text{basis =} \{\mathbf{e_1} , \mathbf{e_2} , \mathbf{e_3} \} \rightarrow \dim(V) = 3 $$

Norm

A function $|| \cdot ||: V \rightarrow \R$ is a norm if and only if it has the following 3 properties.

1. Positive Definiteness. $||x|| \geq 0 \space \text{ and } \space ||x|| =0 \leftrightarrow x = O$