# 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)

## 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.

Created: 2023-06-25 Sun 02:24

Validate