\( A \vec{u}+A^{t} \vec{v} \)
\(=\left(\begin{array}{ccc} 1 & 3 & -1 \\ 2 & 1 & 2 \\ 0 & 1 & 0 \end{array}\right)\cdot \left(\begin{array}{c} 1 \\ 0 \\ -2 \end{array}\right) +\left(\begin{array}{ccc} 1 & 2 & 0 \\ 3 & 1 & 1 \\ -1 & 2 & 0 \end{array}\right)\left(\begin{array}{c} -2 \\ 3 \\ 1 \end{array}\right) \)
\(= \left(\begin{array}{c} 3 \\ -2 \\ 0 \end{array}\right) +\left(\begin{array}{c} 4 \\ -2 \\ 8 \end{array}\right) \)
\(= \left(\begin{array}{c} 7 \\ -4 \\ 8 \end{array}\right) \)