Basis

Remarks

A family $(b_i)_{i \in I}$ is a basis if and only if each $v \in V$ is equal to $\sum_{i \in I} a_i b_i$ for exactly one finitely supported family of scalars $(a_i)_{i \in I}$. The scalars $a_i$ are the coordinates of $v$ in the basis. The standard basis $e_1, \dots, e_n$ is a basis of $F^n$; the vectors $(1, 1)$ and $(1, -1)$ form another basis of $\R^2$. Every vector space has a basis, and all bases of a space have the same cardinality.