Basis

A set of linearly independent vectors \(\{|i\rangle\}\) that spans \(\mathcal{V}\).

Take note of the index: \(\text{span}\{|i\rangle:i=1,\cdots,n\} = \{ \sum_i a^i|i\rangle, \forall a^i\in\mathbb{F} \} = \mathcal{V}\). This implies that every \(|a\rangle\in\mathcal{V}\) may be written as a linear combination of the basis vectors where \(a^i\) are the components of \(|a\rangle\).

Thus, the components, \(\{a^i\}\) are a representation of the vector \(|a\rangle\) in the basis \(\{|i\rangle\}\). \(|a\rangle := \begin{pmatrix}a^1\\a^2\\\cdots\end{pmatrix}\). Where \(:=\) means is represented by in this document.

Incidentally, this implies that once you understand one representation, you understand them all. I.e. there exists an isomorphic map. (Finite dimensional spaces)