1 point A vector space over R is a set V of objects called v

(1 point) A vector space over R is a set V of objects (called vectors) together with two operations, addition and multiplication by scalars (real numbers), that satisfy the following 10 axioms. The axioms must hold for a vectors u, v, w in V and for a scalars a, B in R w 1. (Closed under addition:) The sum of u and v, denoted u v, is in V. 2. (Closed under scalar multiplication:) The scalar multiple of u by a, denoted au is in V. 3. (Addition is commutative u v v u. 4. (Addition is associative:) (u v) w u (v w). 5. (A zero vector exists:) There exists a vector 0 in V such that u 0 u. 6. (Additive inverses exis For each u in V, there exists a v in V such that u +v 0. (We write v -u.) 7. (Scaling by 1 is the identity:) lu u 8. (Scalar multiplication is associative): a(Bu) (aB)u. 9. (Scalar multiplication distributes over vector addition:) a (u v) au av 10. (Scalar addition is distributive:) (a B) au Bu.

Solution

If f and g are 2 vectors in V, then f(x) +g(x) = g(x) +f(x) for all real numbers x since adding the real numbers f(x) and g(x) is a commutative operation. The zero vector in V is the function f given by f(x) = 0 for all x, The additive inverse of the function f in V is a function g that satisfies f(x) +g(x) = 0 for vall real numbers x. The additive inverse of f is the function g(x) = -f(x) for all x. If c is any real number and f and g are two vectors in V, then c(f+g)(x) = c(f(x)+g(x)) = cf(x)+cg(x)