Vector Spaces

Definition

A Vector Space is a Set $\mathbb{V}$ with two operations, "+" and "·"

"+"

The addition map takes 2 Vectors and maps to another vector.

Formally: $+:\mathbb{V} \times \mathbb{V} \rightarrow \mathbb{V}$ $(v,w) \rightarrow v + w$

This is a generalization of the familiar addition and vector addition that you've seen until this point. You can simply add a vector componentwise and receive another vector. However, due to this generalization, you start to pick up other valid "additions" that still work under the Vector Space Axioms.

"·"

The scalar multiplication map takes a vector and a scalar and maps to another vector.

Formally: $\cdot:\mathbb{F} \times \mathbb{V} \rightarrow \mathbb{V}$ $(a,w) \rightarrow a \cdot w$

Where $\mathbb{F}$ is some field often $\mathbb{R}$ or $\mathbb{C}$, and $\mathbb{V}$ is your Vector Space. Similarly, this formalizes the notions of regular multiplication and scalar multiplication that are familiar to you. If you multiply a vector by a scalar, you can simply distribute this multiplication elementwise and end up with another vector.

Properties of a Vector Space

This may be the first formal mathematical "object" you've seen before. These objects are defined by their structure or properties.

The axioms that define a vector space are:

Examples

3 dimensional real space or $\mathbb{R}^3$ is a real vector space, it obeys all above axioms.