Let V and W be vector spaces, and consider the vector space of linear maps L(V, W) with the usual addition and scalar multiplication of functions. We can define a new vector space V × L(V, W): vectors in this space are pairs (v, f) where the first component is a vector v ? V and the second component is a linear map f : V ?? W, addition of pairs is defined as follows:
(v, f) ? (z, g) = (v +V z, f +L g)
where +V is addition in V and +L is addition of functions in L(V, W). Scalar multiplication is defined similarly:
r ? (v, f) = (r ·V v, r ·L f)
where again ·V is scalar multiplication in V and ·L is scalar multiplication of functions in L(V, W). We can define a function
ev : V × L(V, W) ?? W by ev(v, f) = f(v)
Prove that ev is linear.