Let A be the matrix below and define a transformation $T: \mathbb{R}^3 \to \mathbb{R}^3$ by $T(U) = AU.$ For each of the vectors $B$ below, find a vector $U$ such that $T$ maps $U$ to $B$, if possible. Otherwise state that there is no such $U$. $$ \begin{pmatrix} 1 & -3& 2 \\ 2& -4& 4 \\ 3& -8& 6\\ \end{pmatrix} =A $$
a)$$ \begin{pmatrix} 4\\ 6\\ 11\\ \end{pmatrix} =B $$ b)$$ \begin{pmatrix} -3\\ -2\\ -7\\ \end{pmatrix} =B $$
Hint: write $U=(u_1,u_2,u_3)$ and then work out $AU$ in terms of these components. You now have a system of linear equations to solve in the form $AU=B$.