Let (Fun(R, R), +,-) be the real vector space of all functions from R to R. Recall that the

addition of two functions f, g € Fun(R, R) and the multiplication of a function f € Fun(R, R) with a real

scalar A E R are explained by (f+g)(t) = f(t) + g(t) and (Af) (t) = Af(t) for all t € R.

a) Pick any real number q € R. Let V(q,0) = {f € Fun(R, R): f(q) = 0} be the set of all functions

f: R → R with the property that their value at q is equal to zero, i.e. f(q) = 0. Explain why V (q,0)

is a subspace of Fun (R, R).

b) Pick any real number q E R. Let V(q,) = {f € Fun(R, R): f(q) = π} be the set of all functions

f: IR→ R with the property that their value at q is equal to , i.e. f(q) = x. Is V(q,) a subspace of

Fun(R, R)? Explain your answer.

c) Finally, consider the subspace V = Span({f1, f2, f3}) of the vector space (Fun(R, R), +,-) spanned by

the following three functions

fi(t) = cos(t)², f2(t) = sin(t)², f3(t) = 4 for t € R.

Show that B = {f1, f2} is a basis of the vector space V and compute the B-coordinates of the vector

f1 + f2+f3 € V.