In Remark 5.4.27 it is claimed that the set \(S=\left\{\lmatrix{r}1\\1\rmatrix ,\,\lmatrix{r}1\\-1\rmatrix\right\}\) is a basis for \(V=\R^2\text{.}\)
9.
Show that the vectors in \(S\) are linearly independent.
10.
Show \(\lmatrix{r} 4 \\-2\rmatrix \in \R^2\) is in the span of \(S\text{.}\) That is, find \(a\) and \(b\) so that \(a\begin{pmatrix}1\\1 \end{pmatrix} +b\lmatrix{r}1\\-1\rmatrix = \lmatrix{r}4\\-2\rmatrix\text{.}\)
Find \(\begin{pmatrix} a \\b \end{pmatrix} = A^{-1} \lmatrix{r}4\\-2\rmatrix\text{.}\) Does this match the answer you got in Part 5.5.1.10 ? Explain why that makes sense.
13.
Explain how you know \(S\) spans \(\R^2\) thus completing the proof that \(S\) is a basis for \(R^2\text{.}\) In other words, given \(\begin{pmatrix} x \\ y \end{pmatrix} \in \R^2\) explain how to find \(a\) and \(b\) so that \(a\begin{pmatrix}1\\1 \end{pmatrix} +b\lmatrix{r}1\\-1\rmatrix = \begin{pmatrix} x \\ y \end{pmatrix}\text{.}\)
14.
Determine if \(\vec{v}=\lmatrix{r} -1\\-8\\10\rmatrix\) is in the span of the set of vectors \(S=\left\{ \lmatrix{r}-1\\0\\2\rmatrix, \lmatrix{c}1\\2\\0\rmatrix, \lmatrix{r}1\\-1\\1\rmatrix\right\}\text{.}\)
15.
Construct a set of vectors in \(\R^3\) that is linearly independent but not a basis for \(\R^3\text{.}\) Explain / justify your claim.
16.
Construct a set of vectors in \(\R^3\) that spans \(\R^3\) but is not a basis for \(\R^3\text{.}\) Explain / justify your claim.
17.
Find a basis \(\mathcal{A}\) for \(\R^2\) other than the one in Problem 5.5.1.9–13 or \(\left\{\left(\begin{array}{r}\pm1\\0 \end{array} \right),\left(\begin{array}{r}0\\\pm1 \end{array} \right)\right\}\) and prove that \(\mathcal{A}\) is a basis.
