\[ \newcommand{R}{\mathbb{R}} \newcommand{defarrow}{\quad \stackrel{\text{def}}{\Longleftrightarrow} \quad} \]

# Bases and dimension

Let \(X\) be a vector space.

## Hamel Bases

A linearly independent set which generates \(X\) is called a **(Hamel) basis** for \(X\): \[ S \subset X \text{ Hamel basis for } X \defarrow \mathrm{span}(S) = X \quad\text{and}\quad S\, \text{ lin. indep.}\]
Equivalently, \(S\) is a Hamel basis for \(X\) if every vector \(x \in X\) has a *unique and finite* representation \[ x = \sum_{\text{finite}} a_j u_j, \qquad u_j \in S.\] We shall consider only **ordered** Hamel bases, in which case the scalars \(a_j\), called **coordinates**, are well defined.

**Ex.**

- \(\{e_1, \ldots, e_n\}\), with \[ e_j = (0,\ldots,\underbrace{1}_{\text{jth position}},0 \ldots) \] is called the
**standard basis**for \(\R^n\).

- \(\{1,x,x^2, \ldots\}\) is an ordered Hamel basis for \(P(\R)\): every real polynomial can be uniquely expressed as a finite sum, \[p(x) = \sum_{\text{finite}} a_j x^j, \qquad a_j \in \R. \]

## Dimension

If \(X\) has a basis consisting of finitely many vectors, \(X\) is said to be **finite-dimensional**. Else, \(X\) is **infinite-dimensional**.

### › The dimension of any finite-dimensional vector space is unique

All bases of a finite-dimensional vector space have the same number of elements. This number is called the **dimension** of the space.

**Ex.**

- \(\R^n\) has dimension \(n\).
- \(P_n(\R)\), has dimension \(n+1\). (Recall that \(P_n(\R) \cong \R^{n+1}\).)
- \(\mathbb C^n\) has dimension \(n\) when considered as a
*complex*vector space, but \(2n\) when considered a*real*vector space. - The \(l_p\)-, \(BC\)-, and \(L_2\)-spaces are all infinite-dimensional.

### › Any finite-dimensional vector space is isomorphic to Euclidean space

Let \(X\) be a real vector space with basis \(\{e_1, \ldots, e_n\}\). Then \(X \cong \R^n\).^{1)}

**N.b.** In an \(n\)-dimensional vector space, \(m > n\) vectors are linearly dependent.

^{1)}