Vector spaces

A vector space over a field \(F\) is a set of vectors (\(\require{physics} \vb{v} \in V\)) together with two binary operations: the vector addition, \(+\), and the scalar multiplication with an element of the field, \(\cdot\), fulfilling the following axioms:

1. \((V,+)\) is an Abelian group

2. \(V\) is closed with respect to scalar multiplication \[\begin{equation} c \cdot \vb{v} \in V \end{equation}\]

3. \(V\) has an identity element for scalar multiplication \[\begin{equation} 1 \in F: 1 \cdot \vb{v} = \vb{v} \end{equation}\]

4. scalar multiplication is compatible with field multiplication \[\begin{equation} a \cdot (b \cdot \vb{v}) = (ab) \cdot \vb{v} \end{equation}\]

5. scalar multiplication is distributive over vector addition \[\begin{equation} a \cdot (\vb{u} + \vb{v}) = a \cdot \vb{u} + a \cdot \vb{v} \end{equation}\]

6. scalar multiplication is distributive over field addition \[\begin{equation} (a + b) \cdot \vb{u} = a \cdot \vb{u} + b \cdot \vb{u} \end{equation}\]

Every field \(\mathbb{F}\) is, in a sense, a vector space over itself, in which scalar multiplication is replaced by the field multiplication, and vector addition is replaced by field addition.

\(\mathbb{R}^n\), with elements being column matrices of dimension \(n\), together with matrix addition and scalar multiplication, forms a vector space over \(\vb{R}\).

\(\mathbb{R}^{m \times n}\), with elements being the \((m \times n)\)-matrices, together with matrix addition and scalar multiplication, forms a vector space over \(\vb{R}\).