Matroid Induced by Linear Independence in Vector Space is Matroid

Theorem
Let $V$ be a vector space.

Let $S$ be a finite subset of $V$.

Let $\struct{S, \mathscr I}$ be the matroid induced on $S$ by linear independence in $V$.

That is, $\mathscr I$ is the set of linearly independent subsets of $S$.

Then $\struct{S, \mathscr I}$ is a matroid.

Proof
It needs to be shown that $\mathscr I$ satisfies the matroid axioms $(I1)$, $(I2)$ and $(I3)$.

Also see

 * Leigh.Samphier/Sandbox/Matroid Induced by Linear Independence in Abelian Group is Matroid