Section 2.11 Maximal Linearly Independent Sets
Checkpoint 2.11.1.
Is \(\mathbb{S} = \{ \ot{1}{2}{3},\ot{1}{2}{0},\ot{1}{0}{0} \}\) a basis for \(\mathbb{R}^3\text{?}\) Explain.
Checkpoint 2.11.2.
Is \(\mathbb{S} = \{x^2+1, x+1, x-1 \}\) a basis for \(\mathbb{P}_3\text{?}\) Explain.
Checkpoint 2.11.3.
State Theorem 2.10.4 in words.
Theorem 2.11.4.
\(\mathbb{S}\) is a basis for \(\mathbb{L}\) if and only if it is a maximal linearly independent subset of \(\mathbb{L}\text{.}\) In other words, \(\mathbb{S}\) is a basis for \(\mathbb{L}\) if and only if \(\mathbb{S}\) is linearly independent and not a proper subset of any other linearly independent set.