Section 2.14 The Replacement Lemma
Checkpoint 2.14.1.
Is the set \(\{1, x, x^2, x^3, \ldots,x^n, \ldots\}\) a basis for C[0,1]?
Checkpoint 2.14.2.
Consider the set \(\mathbb{S}=\{\ot{1}{0}{1}, \ot{0}{1}{0} \}\) which spans some subspace \(\mathbb{L} \subseteq \mathbb{R}^3\text{.}\) Notice \(\ot{2}{3}{2}\) is a linear combination of the points in \(\mathbb{S}\) since \(\ot{2}{3}{2} =2\ot{1}{0}{1}+ 3\ot{0}{1}{0}\text{.}\) Consider the set \(\mathbb{S}' = \{ \ot{1}{0}{1}, \ot{2}{3}{2} \}\text{.}\) Show how the point \(\ot{0}{1}{0}\) can be written as a linear combination of the points in \(\mathbb{S}'\text{?}\) Does \(\mathbb{S}'\) also span \(\mathbb{L}\text{?}\) Explain.
Checkpoint 2.14.3.
If \(\mathbf{Q} = 2\mathbf{P}_1+3\mathbf{P}_2 + 0\mathbf{P}_3 + 4\mathbf{P}_4 + -1\mathbf{P}_5 + 0\mathbf{P}_6\) which \(\mathbf{P}\)'s can be written as linear combinations \(\mathbf{Q}\) and the other \(\mathbf{P}\)'s?
Theorem 2.14.4. Replacement Lemma.
Suppose that \(\mathbb{S}\) spans \(\mathbb{L}\text{,}\) \(\mathbf{Q} \in \mathbb{L}\) and \(\mathbf{P}\) is a point of \(\mathbb{S}\) such that when \(\mathbf{Q}\) is written as a linear combination of points of \(\mathbb{S}\text{,}\) the coefficient of \(\mathbf{P}\) is not zero. If \(\mathbb{S}'\) is the set obtained from \(\mathbb{S}\) by replacing \(\mathbf{P}\) with \(\mathbf{Q}\text{,}\) then \(\mathbb{S}'\) also spans \(\mathbb{L}\text{.}\)