Section 2.12 Reducing a Spanning Set
Checkpoint 2.12.1.
Show how to write the point \(\mathbf{A} = \ot{a}{b}{c}\) as a linear combination of the points in \(\mathbb{S} = \{ \ot{1}{0}{-1},\ot{1}{2}{3},\ot{3}{2}{1},\ot{0}{1}{0} \}\text{.}\) Are all the points in \(\mathbb{S}\) necessary?
Checkpoint 2.12.2.
Find a subset of \(\mathbb{S} = \{ \ot{1}{0}{-1},\ot{1}{2}{3},\ot{3}{2}{1},\ot{0}{1}{0} \}\) that spans \(\mathbb{R}^3\text{.}\)
Checkpoint 2.12.3.
Find a subset of \(\mathbb{S} = \{ \ot{1}{0}{-1},\ot{1}{2}{3},\ot{3}{2}{1},\ot{0}{1}{0} \}\) that does not span \(\mathbb{R}^3\text{.}\)
Theorem 2.12.4.
If \(\mathbb{S}\) spans \(\mathbb{L}\) and \(\mathbf{P}\) is a point of \(\mathbb{S}\) such that \(\mathbf{P} \in span(\mathbb{S}-\{\mathbf{P}\})\) then \(\mathbb{S}-\{\mathbf{P}\}\) also spans \(\mathbb{L}\text{.}\)