\[ \newcommand{R}{\mathbb{R}} \newcommand{defarrow}{\quad \stackrel{\text{def}}{\Longleftrightarrow} \quad} \]
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.
- \(\{e_1, \ldots, e_n\}\), 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.
- \(\R^n\) 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.