Section 2.13 Minimal Spanning Sets
Checkpoint 2.13.1.
Find a basis for \(\mathbb{R}^n\text{.}\) Now find a different basis for \(\mathbb{R}^n\text{.}\)
Checkpoint 2.13.2.
Find two different bases for \(\mathbb{P}_n\text{.}\) Find a basis for \(\mathbb{P}\text{.}\)
Checkpoint 2.13.3.
State Theorem 2.12.4 in words.
Theorem 2.13.4.
\(\mathbb{S}\) is a basis for \(\mathbb{L}\) if and only if \(\mathbb{S}\) is a minimal spanning set for \(\mathbb{L}\text{.}\) In other words \(\mathbb{S}\) is a basis for \(\mathbb{L}\) if and only if \(\mathbb{S}\) spans \(\mathbb{L}\) and no proper subset of \(\mathbb{S}\) spans \(\mathbb{L}\text{.}\)