$ U = \op{span } \left\{ \mat{-3\\-2\\3\\-2}, \mat{2\\3\\-1\\-1} \right\} $ Write an algabraic specification of the [[../02 Areas/Math/Span|Span]] of this [[../02 Areas/Math/Linear Subspace|Linear Subspace]]. $\begin{align} x_{1}\mat{-3\\-2\\3\\-2} + x_{2}\mat{2\\3\\-1\\-1} &= \underbrace{ \mat{-3 & 2 \\-2 & 3\\3 & -1\\-2 & -1}\mat{x_{1}\\x_{2}}}_{A}\\ U &= \op{Col}(A) \\ &\sim \augmented{cc|c}{ -3 & 2 & b_{1} \\ -2 & 3 & b_{2} \\ 3 & -1 & b_{3} \\ -2 & -1 & b_{4} }\\ &\sim \augmented{cc|c}{ -3 & 2 & b_{1} \\ -2 & 3 & b_{2} \\ 0 & 1 & b_{1}+ b_{3} \\ 0 & -4 & -b_{2}+b_{4} }\\ &\sim \augmented{cc|c}{ -2 & 2 & b_{1} \\ 0 & \frac{5}{3} & b_{2} \\ 0 & 1 & b_{1}+ b_{3} \\ 0 & 0 & 4b_{1} -b_{2} + 4b_{3} +b_{4} }\\ &\sim \augmented{cc|c}{ -3 & 2 & b_{1} \\ 0 & \frac{5}{3} & b_{2} \\ 0 & 0 & \frac{7}{5}b_{1} -\frac{3}{5}b_{2}+b_{3} \\ 0 & 0 & 4b_{1} -b_{2} + 4b_{3} +b_{4} } \end{align} $ $\begin{align} U &= \setbuild{ \vec b \in \R^4 }{ \begin{split} \frac{7}{5} b_{1} - \frac{3}{5} b_{2} + b_{3} &= 0 \\ 4b_{1} - b_{2} + 4b_{3} + b_{4} &= 0 \end{split} } \end{align} $ *Note*: alternative solution is to get the [[../02 Areas/Math/Reduced Row Echelon Form|RREF]] of $\augmented{c|c}{A & I}$.