(Solved): Let S be the set given in Exercise 14. For each vector given below, determine whether the vecto ...

Let S be the set given in Exercise 14. For each vector given below, determine whether the vector is in Sp(S ). Express those vectors that are in Sp(S ) as a linear combination of v and w.

reference for v and w

question 20 is what needs to be solved using Sp(S) = {v,w}

Exercises 12-19 refer to the vectors in Eq. (15). \[ \begin{array}{l} \mathbf{v}=\left[\begin{array}{l} 1 \\ 2 \\ 0 \end{array}\right], \quad \mathbf{w}=\left[\begin{array}{r} 0 \\ -1 \\ 1 \end{array}\right], \quad \mathbf{x}=\left[\begin{array}{r} 1 \\ 1 \\ -1 \end{array}\right], \\ \mathbf{y}=\left[\begin{array}{r} -2 \\ -2 \\ 2 \end{array}\right], \quad \mathbf{z}=\left[\begin{array}{l} 1 \\ 0 \\ 2 \end{array}\right] \end{array} \] 20. Let \( S \) be the set given in Exercise 14. For each vector given below, determine whether the vector is in \( \operatorname{Sp}(S) \). Express those vectors that are in \( \operatorname{Sp}(S) \) as a linear combination of \( \mathbf{v} \) and \( \mathbf{w} \). a) \( \left[\begin{array}{l}1 \\ 1 \\ 1\end{array}\right] \) b) \( \left[\begin{array}{r}1 \\ 1 \\ -1\end{array}\right] \) c) \( \left[\begin{array}{l}1 \\ 2 \\ 0\end{array}\right] \) d) \( \left[\begin{array}{l}2 \\ 3 \\ 1\end{array}\right] \) e) \( \left[\begin{array}{r}-1 \\ 2 \\ 4\end{array}\right] \) f) \( \left[\begin{array}{l}1 \\ 1 \\ 3\end{array}\right] \)