0.1 Vector Space Axioms
In mathematics and physics, [[vector|vector]] is a term that refers informally to some quantities that cannot be expressed by a single number (a scalar), or to elements of some vector spaces.
Given some field \(F\) we can define a vector space (also known as a linear space) which is a collection of vectors with elements from field \(F\) that satisfies the vector space axioms.
| Axiom | Statement |
|---|---|
| Associativity of vector addition | u + (v + w) = (u + v) + w |
| Commutativity of vector addition | u + v = v + u |
| Identity element of vector addition | There exists an element 0 ∈ V, called the zero vector, such that v + 0 = v for all v ∈ V. |
| Inverse elements of vector addition | For every \(\vec{v} \in \mathcal{V}\), there exists an element −v ∈ V, called the additive inverse of v, such that v + (−v) = 0. |
| Compatibility of scalar multiplication with field multiplication | \(a(b \vec{v}) = (ab)\vec{v}\) |
| Identity element of scalar multiplication | \(1\vec{v}=\vec{v}\), where 1 denotes the multiplicative identity in F. |
| Distributivity of scalar multiplication with respect to vector addition | a(u + v) = au + av |
| Distributivity of scalar multiplication with respect to field addition | (a + b)v = av + bv |
0.2 Properties of Vector Spaces
A vector space \((\mathcal{V}, F)\) is linear, that is for all \(v_{1}, v_{2} \in \mathcal{V}\) and \(\alpha,\beta \in F\), the linear combination \[ \alpha v_{1}+\beta v_{2}\in\mathcal{V}. \]