Basis

A set of linearly independent vectors ${| i ⟩}$ that spans $V$ .

Take note of the index: $span {| i ⟩ : i = 1, \dots, n} = {\sum_{i} a^{i} | i ⟩, \forall a^{i} \in F} = V$ . This implies that every $| a ⟩ \in V$ may be written as a linear combination of the basis vectors where $a^{i}$ are the components of $| a ⟩$ .

Thus, the components, ${a^{i}}$ are a representation of the vector $| a ⟩$ in the basis ${| i ⟩}$ . $| a ⟩ := (\begin{matrix} a^{1} \\ a^{2} \\ \dots \end{matrix})$ . 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)

Theorem of Uniqueness of Components

The ${a^{i}}$ constitute a unique set.

Proof. If there was another set, then their difference gives a zero vector, which violates the linear independence of the basis.

Definition

A basis for a vector space is any spanning set of linear independent vectors of the space.