MA22020: Advanced linear algebra

1.3 Bases

Definitions. Let \(\lst {v}1n\) be a list of vectors in a vector space \(V\).
- (1) The span of \(\lst {v}1n\) is
  \(\seteqnumber{0}{1.}{0}\)
  \begin{equation*} \Span {\lst {v}1n}:=\set {\lc \lambda {v}1n\st \lambda _i\in \F , 1\leq i\leq n}\leq V. \end{equation*}
- (2) \(\lst {v}1n\) span \(V\) (or are a spanning list for \(V\)) if \(\Span {\lst {v}1n}=V\).
- (3) \(\lst {v}1n\) are linearly independent if, whenever \(\lc \lambda {v}1n=0\), then each \(\lambda _i=0\), \(1\leq i\leq n\), and linearly dependent otherwise.
- (4) \(\lst {v}1n\) is a basis for \(V\) if they are linearly independent and span \(V\).

Remark. Notice that any re-ordering of a basis is also a (different) basis. Example: if \(v_1,v_2v_3\) is a basis, so is \(v_2,v_1,v_3\) and so on.

Definition. A vector space is finite-dimensional if it admits a finite list of vectors as basis and infinite-dimensional otherwise.

If \(V\) is finite-dimensional, the dimension of \(V\), \(\dim V\), is the number of vectors in a (any) basis of \(V\).

Terminology. Let \(\lst {v}1n\) be a list of vectors.
- (1) A vector of the form \(\lc \lambda {v}1n\) is called a linear combination of the \(v_i\).
- (2) An equation of the form \(\lc \lambda {v}1n=0\) is called a linear relation on the \(v_i\).

Example. Some lucky vector spaces come with a natural choice of basis. For instance, define \(e_i:=(0,\dots ,1,\dots ,0)\in \F ^n\), \(\bw 1in\) with a single \(1\) in the \(i\)-th place and zeros elsewhere. Then \(\lst {e}1n\) is a basis of \(\F ^n\) called the standard basis

1.3.1 Useful facts

A very useful fact about bases that we shall use many times was proved in Algebra 1B:

Proposition 1.1 (Algebra 1B, Corollary 1.5.7). Any linearly independent list of vectors in a finite-dimensional vector space can be extended to a basis.

Here is another helpful result :

Proposition 1.2 (Algebra 1B, Corollary 1.5.6). Let \(V\) be a finite-dimensional vector space and \(U\leq V\). Then
\(\seteqnumber{0}{1.}{0}\)
\begin{equation*} \dim U\leq \dim V \end{equation*}

with equality if and only if \(U=V\).