Section 2.17 Properties of Spanning Sets
Checkpoint 2.17.1.
Show that no finite set spans C[0,1].
Checkpoint 2.17.2.
Describe how to reduce a spanning set to get a basis. Will this always work?
Checkpoint 2.17.3.
Use negation to rewrite Theorem 18.
Theorem 2.17.4.
If \(\mathbb{L}\) has a basis \(\mathbb{B} = \{ \mathbf{P}_1, \mathbf{P}_2,\ldots,\mathbf{P}_n \}\) with a finite number of points, then the following hold:
No spanning set contains fewer than \(n\) points.
Every spanning set with \(n\) points is a basis.
Every finite spanning set contains a basis.
Theorem 2.16.4 and Theorem 2.17.4 together give the result that if a linear space has a basis with \(n\) points, then every basis must have \(n\) points. This observation leads to the following important definition.
Definition 2.17.5.
If a linear space \(\mathbb{L}\) has a basis with \(n\) elements then \(\mathbb{L}\) is called an n dimensional linear space. If there is no finite set that forms a basis for \(\mathbb{L}\text{,}\) \(\mathbb{L}\) is said to be an infinite dimensional linear space.