Section 2.16 Properties of Linearly Independent Sets
Checkpoint 2.16.1.
Is there an infinite subset in C[0,1] which is linearly independent? To demonstrate such a set one option is to use pictures and establish a pattern.
Checkpoint 2.16.2.
Describe how to extend a linearly independent set to get a basis. Will this always work?
Checkpoint 2.16.3.
Use negation to rewrite Theorem 16.
Theorem 2.16.4.
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 linearly independent set contains more than \(n\) points.
Every linearly independent set with \(n\) points is a basis.
Every linearly independent set is contained in a basis.