Let V be \mathbb R-vector space,
\mathcal F: V \to \mathbb R
a linear map and
v, u \in V with \mathcal F(Vv) = A and
\mathcal F(Uu) = B.
Determine \mathcal F(Cv + Du).
\mathcal F(Cv + Du)
=
C*A/V+D*B/U
Using the linear property, we have \mathcal F(Cv + Du) = C {\color{red}\mathcal F(v)} +
D{\color{blue}\mathcal F(u)}.
In the same manner \mathcal F(Vv) = V {\color{red}\mathcal F(v)} = A and
\mathcal F(Uu) = U {\color{blue}\mathcal F(u)} = B.
Expanding and substituting gives: \mathcal F(Cv + Du) = C {\color{red}\mathcal F(v)} +
D{\color{blue}\mathcal F(u)}=
fractionReduce(C*A*U+D*B*V,U*V).