Vector Equations

1. Determine if \( \vec{b} = \left[\begin{array}{r}-5 \\ 11 \\ -7 \end{array}\right]\) is a linear combination of the vectors \( \vec{a_1} = \left[\begin{array}{r}1 \\ -2 \\ 2 \end{array}\right], \vec{a_2} = \left[\begin{array}{r} 0 \\ 5 \\ 5 \end{array}\right], \vec{a_3} = \left[\begin{array}{r} 2 \\ 0 \\ 8 \end{array}\right]\).     solution

2. Let \( \vec{v_1} = \left[\begin{array}{r}1 \\ 0 \\ -2 \end{array}\right],  \vec{v_2} = \left[\begin{array}{r} -3 \\ 1 \\ 8 \end{array}\right], \) and \( \vec{x} = \left[\begin{array}{r}a \\ -5 \\ -3 \end{array}\right]\). For what value(s) of \(a\) is \(\vec{x}\) in the plane generated by \(\vec{v_1}\) and \(\vec{v_2}\)? solution

3. Determine if \( \vec{b} = \left[\begin{array}{r}11 \\ -5 \\ 9 \end{array}\right]\) is in the span\(\{  \vec{a_1}, \vec{a_2} , \vec{a_3}  \}\). If yes, express \(\vec{b} \) as a linear combination of the vectors \(\vec{a_1}, \vec{a_2}\) and \( \vec{a_3}  \).

\(\vec{a_1} = \left[\begin{array}{r} 11 \\ -5 \\ 9 \end{array}\right], \vec{a_2} = \left[\begin{array}{r} 1 \\ 0 \\ 1 \end{array}\right], \vec{a_3} = \left[\begin{array}{r} -2 \\ 3 \\ -2 \end{array}\right], \vec{b} = \left[\begin{array}{r} -6 \\ 7 \\ 5 \end{array}\right] \)

solution

4. Let \( A=\left[\begin{array}{rrr}
{1} & {0} & {2} \\
{0} & {3} & {3} \\
{-2} & {6} & {2}
\end{array}\right] \), let \(\vec{b}=\left[\begin{array}{r}
{4} \\
{1} \\
{-4}
\end{array}\right] \), and let \(W\) be the set of all linear combinations of columns of \(A\).
(a) Is \( \vec{b} \) in \(W\)?
(b) Show that the second column of \(A\) is in \(W\).
solution