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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).Then \displaystyle{ f \left( \begin{array}{r} -2 \\ 6 \end{array} \right) = }.
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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).Then \displaystyle{ f ( 3 {\vec{e}}_4 + 7 {\vec{e}}_7 ) - f ( 5 {\vec{e}}_8 + 4 {\vec{e}}_7 )= }
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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).Then -4 T({\vec{v}}_1) + T(4 {\vec{v}}_2 + 5 {\vec{v}}_3) =