Determine where the following subsets are vector spaces or not?

Question

Determine whether the following subsets of C^2 (R) are vector spaces or not (that is, whether they're subspaces of C^2(R) or not).

a) The set of functions, f, satisfying f(1) = f(2) - 1.

b) The set of functions, f, satisfying f(1) = f(2).

Answer 1

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The second is, the first is not.

a) To show it's not a subspace, you can note that it doesn't contain the zero vector--this is the function f(x) = 0 for all x. Alternatively, suppose f is in your set. Note then that 2f is not---it's not closed under scalar multiplication. See that

2f(1) = 2(f(2) - 1) = 2f(2) - 2 ≠ 2f(2) - 1.

b) The zero vector would satisfy this condition, so there's no problem there. Suppose f and g are in your set and let k be any scalar. Remember that f(1) = f(2) and g(1) = g(2) by our hypothesis. Observe that

(f + g)(1) = f(1) + g(1) = f(2) + g(2) = (f + g)(2), and

(kf)(1) = kf(1) = kf(2) = (kf)(2).

So you can see that the necessary property is satisfied by vector sums and scalar multiples. That is, your set is closed under both operations.