Linearly Independent Set is Basis iff of Same Cardinality as Dimension

Theorem
Let $E$ be a vector space of $n$ dimensions.

Let $H$ be a linearly independent subset of $E$.

$H$ is a basis for $E$ it contains exactly $n$ elements.

Proof
By hypothesis, let $H$ be a linearly independent subset of $E$

Necessary Condition
Let $H$ be a basis for $E$.

By definition of dimension of vector space, a basis for $E$ contains exactly $n$ elements.

By Bases of Finitely Generated Vector Space have Equal Cardinality, it follows that $H$ also contains exactly $n$ elements.

Sufficient Condition
Let $H$ contain exactly $n$ elements.

By Sufficient Conditions for Basis of Finite Dimensional Vector Space $H$ is itself a basis for $E$.