Let V be the set of functions f RR. For any two functions f,g in V, define the s

Question

Let V be the set of functions f RR. For any two functions f,g in V, define the sumf +g to be the function given by (f+ g)(x)-f(x) + g(x) for all real numbers x. For any real number c and any function f in V, define scalar multiplication cf by (cf)(x) cf(x) for all real numbers x. Answer the following questions as partial verification that V is a vector space. (Addition is commutative:) Letf and g be any vectors in V. Then/x) + g(x) = f(x) and g(x) is a commutative operation. (A zero vector exists:) The zero vector in V is the function f given by fx) for all real numbers x since adding the real numbers for all x. (Additive inverses exist:) The additive inverse of the function f in V is a function g that satisfies f(x) + g(x) 0 for all real numbers x. The additive inverse of f is the function gCx)- for all x. (Scalar multiplication distributes over vector addition:) If c is any real number andf and g are two vectors in V, then cr + g)(x) = c(f(x) + g(x))-

Explanation / Answer

1.Addition is commutative i.e f(x) + g(x) = g(x) + f(x)

2.The zero vector is f(x)=0 for all x. Since by adding f(x) = 0 in any vector gives the same vector.

3.The additive inverse of f is the function g(x) = -f(x) for all x as

f(x) + g(x) = f(x) - f(x) =0

4.Scalar multiplication distributes over addition i.e

c(f + g)(x) = c(f(x) + g(x)) =  cf(x) + cg(x)

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