system of generators

System of Generators

$B=\{{\mathbf{e}}_1,{\mathbf{e}}_2,\dots ,{\mathbf{e}}_n\}\subset V$ is called a system of generator of the vector space $V$ if every vector $\mathbf{v}\in V$ can be written as a linear combintation of the vectors in $B$:

$$\mathbf{v}={\alpha }_1{\mathbf{e}}_1+{\alpha }_2{\mathbf{e}}_2+\cdots +{\alpha }_n{\mathbf{e}}_n,{\alpha }_i\in F$$

where $F$ is the underlying field of the vector space $V$.

A system of generators may contain redundant elements, i.e. some vectors in $B$ can be expressed as a linear combination of the others.