Let V be \mathbb R-vector space
\mathcal F: V \to \mathbb R
a linear map and
v, u \in V with {\color{red}\mathcal F(v) = A} and
{\color{blue}\mathcal F(u) = B}.
Determine \mathcal F(Cv + Du).
\mathcal F(Cv + Du)
=
C*A+D*B
Using the linear property, we have \mathcal F(Cv + Du) = C {\color{red}\mathcal F(v)} +
D{\color{blue}\mathcal F(u)}.
Substituting {\color{red}\mathcal F(v) = A} and
{\color{blue}\mathcal F(u) = B} yields
\mathcal F(Cv + Du) = C {\color{red}\mathcal F(v)} +
D{\color{blue}\mathcal F(u)}=
C*A+D*B.