(a) Suppose f : \mathbb{R}^{2} \to \mathbb{R}^{3} is a linear transformation such that f \left( \begin{array}{c} 1 \\ 0 \end{array} \right) = \left( \begin{array}{r} 2 \\ -3 \\ -3 \end{array} \right), \quad f \left( \begin{array}{c} 0 \\ 1 \end{array} \right) = \left( \begin{array}{r} -3 \\ -4 \\ 3 \end{array} \right).Compute \displaystyle{ f \left( \begin{array}{r} -2 \\ 6 \end{array} \right)}.

(b) Suppose f : \mathbb{R}^{12} \to \mathbb{R}^{2} is a linear transformation such that f \left( \begin{array}{c} {\vec{e}}_4 \end{array} \right) = \left( \begin{array}{r} 2 \\ -1 \end{array} \right), \quad f \left( \begin{array}{c} {\vec{e}}_7 \end{array} \right) = \left( \begin{array}{r} -2 \\ 5 \end{array} \right), \quad f \left( \begin{array}{c} {\vec{e}}_8 \end{array} \right) = \left( \begin{array}{r} 1 \\ -1 \end{array} \right).Compute \displaystyle{ f ( 3 {\vec{e}}_4 + 7 {\vec{e}}_7 ) - f ( 5 {\vec{e}}_8 + 4 {\vec{e}}_7 )}.

(c) Let V be a vector space and let {\vec{v}}_1, {\vec{v}}_2, {\vec{v}}_3 \in V. Suppose T : V \to \mathbb{R}^{2} is a linear transformation such that T({\vec{v}}_1) = \left( \begin{array}{r} -5 \\ 4 \end{array} \right), \quad T({\vec{v}}_2) = \left( \begin{array}{r} -3 \\ -1 \end{array} \right), \quad T({\vec{v}}_3) = \left( \begin{array}{r} -5 \\ 5 \end{array} \right).Compute -4 T({\vec{v}}_1) + T(4 {\vec{v}}_2 + 5 {\vec{v}}_3).