Thread: Vector space of functions

Ive been working on this problem for hours but i have yet figure out how to solve it.

Let V be the vector space of functions f : R -> R. Show that

W = { F(x) : F(1) = 0},
the set of all functions whose value at 1 is 0, is a subspace of V,

What ideas have you had so far?

show that F(1) = 0 is a zero vector or apply vector addition, or multiply by a constant, i'm not really so sure on how to approach this problem.

How do you normally show that a subspace of a vector space is itself a vector space?

A subspace is simply a subset that is, itself a complete vector space. Since you are using the same addition and scalar multiplication as in the vector space, you do not have to show things like commutativity of addition, associativity, etc.

You need to show three things:
1) The set is non-empty (which you can often do by showing that the 0 vector is in the set).
2) The set is closed under addition of vectors (show that the sum of two vectors in the set is still in the set).
3) The set is closed under scalar multiplication (show that any multiple of a vector in the set is still in the set).