\begin{exercise}{V2}{Linear combinations}{0003}
\begin{exerciseStatement}
Consider the following statement.
\begin{itemize}
\item The vector \( \left[\begin{array}{c}
7 \\
10 \\
16 \\
7
\end{array}\right] \)is not a linear combination of the vectors \( \left[\begin{array}{c}
2 \\
2 \\
2 \\
-1
\end{array}\right] , \left[\begin{array}{c}
-4 \\
-2 \\
-4 \\
1
\end{array}\right] , \left[\begin{array}{c}
-1 \\
-2 \\
-4 \\
-3
\end{array}\right] , \text{ and } \left[\begin{array}{c}
-1 \\
2 \\
-3 \\
-2
\end{array}\right] \).
\end{itemize}
\begin{enumerate}[(a)]
\item Write an equivalent statement using a vector equation.
\item Explain why your statement is true or false.
\end{enumerate}
\end{exerciseStatement}
\begin{exerciseAnswer}\[\operatorname{RREF} \left[\begin{array}{cccc|c}
2 & -4 & -1 & -1 & 7 \\
2 & -2 & -2 & 2 & 10 \\
2 & -4 & -4 & -3 & 16 \\
-1 & 1 & -3 & -2 & 7
\end{array}\right] = \left[\begin{array}{cccc|c}
1 & 0 & 0 & 0 & 2 \\
0 & 1 & 0 & 0 & 0 \\
0 & 0 & 1 & 0 & -3 \\
0 & 0 & 0 & 1 & 0
\end{array}\right] \]
\begin{enumerate}[(a)]
\item The vector equation \( x_{1} \left[\begin{array}{c}
2 \\
2 \\
2 \\
-1
\end{array}\right] + x_{2} \left[\begin{array}{c}
-4 \\
-2 \\
-4 \\
1
\end{array}\right] + x_{3} \left[\begin{array}{c}
-1 \\
-2 \\
-4 \\
-3
\end{array}\right] + x_{4} \left[\begin{array}{c}
-1 \\
2 \\
-3 \\
-2
\end{array}\right] = \left[\begin{array}{c}
7 \\
10 \\
16 \\
7
\end{array}\right] \)has no solutions.
\item
\( \left[\begin{array}{c}
7 \\
10 \\
16 \\
7
\end{array}\right] \) is a linear combination of the vectors \( \left[\begin{array}{c}
2 \\
2 \\
2 \\
-1
\end{array}\right] , \left[\begin{array}{c}
-4 \\
-2 \\
-4 \\
1
\end{array}\right] , \left[\begin{array}{c}
-1 \\
-2 \\
-4 \\
-3
\end{array}\right] , \text{ and } \left[\begin{array}{c}
-1 \\
2 \\
-3 \\
-2
\end{array}\right] \).
\end{enumerate}
\end{exerciseAnswer}
\end{exercise}
