Let $U \subset \R^5$ be the [[../02 Areas/Math/Linear Subspace|Linear Subspace]]: $ U = \op{span}\pa{ \mat{1\\0\\-1\\3\\1}, \mat{2\\1\\2\\0\\2}, \mat{0\\1\\3\\1\\0}} $ Write an algebraic specification of $U$. $ \let A = \mat{ 1&2&0 \\ 0&1& 1 \\ -1&2&3 \\ 3&0&1 \\ 1&2&0 } $ $ U = \op{col}(A) $ $\begin{align} & \augmented{ccc|c}{ 1&2&0 &b_{1}\\ 0&1& 1& b_{2}\\ -1&2&3&b_{3} \\ 3&0&1& b_{4}\\ 1&2&0& b_{5}} \\ \cdots \sim & \mat{ 1 & 2 & 1 & b_{1} \\ 0&1&1 & b_{2} \\ 0&0&-1 & b_{1}-4b_{2}+b_{3} \\ 0&0&0 & 4b_{1}-22b_{2}+7b_{3}+b_{4} \\ 0&0&0& -b_{1}+b_{5} } \end{align} $ $\huge U = \left\{ \vec b \in \R^3 \left| \,\, \begin{split} 4b_{1}-22b_{2}+7b_{3}+b_{4} &= 0 \\ -b_{1}+b_{5} &=0 \end{split} \right. \right\} $ $\huge \op{\mathrm{Im}}(T) = \set{ \vec b \in \R^3 \mid -b_{1}+3b_{2}+b_{3} = 0 } $