In linear algebra, how do you verify that F is a vector space?

Answer 1

You have to be more specific than this. Usually, F represents the base field for a vector space rather than the vector space itself. Anyway, the steps to verify that F is a vector space:
1) Show that the sum of two things in F is again in F
2) Show that anything in F multiplied by a scalar is again in F.
3) Show that (x+y)+z=x+(y+z) for all x,y, and z in F,
4) Show that 1*x=x for all x in F,
5) Show that is a and b are scales and x is in F, then (a+b)x=ax +bx
6) Show that is a is a scalar and x and y are in F, the a(x+y)=ax +ay

Answer 2

The first answer is close to correct, but it isn't quite right (as is typical with mathematician). First, one needn't reserve the letter F to represent the field of scalars--only inexperienced mathematicians get hung up on such minutiae. Also, he forgot to mention several important axioms that need to be verified.

--You need to show the existence of an additive identity for the vector space F, that is, you need to show that there exists an element z in F such that for every x in F, then

z+x = x+z = x.

--You need to show that each element of F has an additive inverse, that is, for every x in F, there exists an element -x in F such that

x + (-x) = (-x) + x = z, where z is an additive identity.

--You need to show that the addition in the vector space needs to be commutative, that is,

for every pair x, y in F, then x+y = y+x.

Otherwise, his answer is (amateurishly) 66% correct.