It's an affine subspace (in this case a linear manifold not containing the origin) in some vector space.

You need to show that the space is closed under affine combinations (i.e., linear combinations with the given restriction on the scalars).

Another way to look at it is that the space is the affine span of {a,b,c}.

To get started, you might look at what Wiki has to offer regarding "affine subspaces".