Question

Show that the set of all vectors of the form (a,-a,c) forms a subspace of R3.

Answer 1

One way to think of this is this subset is
\[ S = \{ a(1,-1,0) + c(0,0,1) | \ a, c \in \mathbb{R} \} \]

It is easy to see then that S is in fact a subspace of R^3 spanned by basis vectors (1,-1,0) and (0,0,1).

Answer 2

But I'm guessing that you'll need to prove it using the definition of sub-space.

So what you have to show is that
1. S contains the zero vector (0,0,0)
2. For any vectors v and w in S, then v + w is in S; and
3. For any vector v in S and any real number x in R, then xv is in S.