  In this paper, we prove that the world of near-vector spaces allows us to
work with non-linear problems and yet, gives access to most of the tools linear
algebra has to offer. We establish some fundamental results for near-vector
spaces toward extending classical linear algebra to near-linear algebra. In the
present paper, we finalize the algebraic proof that any non-empty $F$-subspace
stable under addition and scalar multiplication is an $F$-subspace. We
demonstrate that any quotient of a near-vector space by an $F$-subspace is a
near-vector space and the First Isomorphism Theorem for near-vector spaces. In
doing this, we obtain fundamental descriptions of the span. Defining linear
independence outside the quasi-kernel, we prove that near-vector spaces are
characterized in terms of the existence of a scalar basis, and we obtain a new
important notion of basis.
