determine if the following sets of vectors are vector spaces:

V = {(x y z) from R^3 | ax + by + cz = 0, a, b, c from reals and not all 0}

 

Help would be much appreciated

Two options for you:

1) The mathematical one: Look up the definition of a vector space (group under addition and well defined multiplication by a scalar, maybe others) and check if your vectors meet the conditions.

2) The geometrical one: If you´re a bit familiar with R³ you might notice that the condition ax + by + cz = 0 is an equation for a plane.

Yes, it is a vector Space indeed. A vector space is a set of vectors associated to a group of scalars which fulfills the next conditions:

 

a(V1+V2) = aV1 + aV2

V1*V2 = V2*V1 ; Vi vectors, a scalar

 

Now, R3 is a Vector field, and the quation you present, which is the equation of an arbitrary plane, is a subspace of R3. In fact, it is the space of all R3 planes.