algebra (proof of subspace)?

Question

let A be the set of all f in C^2 (negative infinity,infinity) such that f"(x)+f(x) = 0 for all x in R

prove that A is the subspace of C^2(negative infinity, infinity)

Answer 1

Well, if f and g are in A then, for every real x we have (f+g)''(x) + (f+g)(x) = f''(x) + g''(x) + f(x) + g(x) =f''(x) + f(x) + g''(x) + + g(x) = 0 + 0 = 0, which shows f + g is in A.

For every real a , (a*f)''(x) + (a*f)(x) = a (f"(x)+f(x)) = a * 0 = 0, which shows the function a*f is in A.

Therefore , A is a subspace of C^2, over the reals.