For each of the following functions decide whether or not the function is a linear transformation. Prove your answer in each case:
(a) F: V3(R) --> V2(R) defined by F(a1, a2, a3) = (2a1 - a2, 3a3)
(b) G: V3(R) --> V2(R) defined by G(a1, a2, a3) = (a1 - a2 + a3, a1 + a2a3)
Please find the solutions/explanations attached herewith.
I hope it will help.
Let u = (u1, u2, u3) and v = (v1, v2, v3)
(u + v) = (u1+v1, u2+v2, u3+v3)
F(u + v)
= F(u1+v1, u2+v2, u3+v3)
= (2(u1+v1)-(u2+v2), 3(u3+v3))
F(u) = (2u1 -u2, ...
A linear transformation is embodied.