Is the subset of F with the property that f0 0 a vector spa

Is the subset of F with the property that f(0) = 0 a vector space?

Is the subset of F with the property that f(x) = f(x) for all x a vector space?

Solution

The subset of F with property that f(0) = 0 is not a vector space because in this case f(x) doesn\'t satisfy closeness under scalar multiplication

The subset of F with property f(-x) = -f(x) is a vector space as in this case f(x) satisfies both vector addition and scalar multiplication properties.