Results concerning Generators and Bases of Vector Spaces

Theorem

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

Let $G$ be a generator for $E$.

Then $G$ has the following properties:

Cardinality of Generator of Vector Space is not Less than Dimension

Let $V$ be a vector space over a field $F$.

Let $\BB$ be a generator for $V$ containing $m$ elements.

Then:

$\map {\dim_F} V \le m$

where $\map {\dim_F} V$ is the dimension of $V$.

Generator of Vector Space Contains Basis

$G$ contains a basis for $E$.

Generator of Vector Space is Basis iff Cardinality equals Dimension

$G$ is a basis for $E$ if and only if $\card G = n$.

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

Then $H$ has the following properties:

Cardinality of Linearly Independent Set is No Greater than Dimension

$H$ has at most $n$ elements.

Finite Dimensional Case

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

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

There exists a basis $B$ for $E$ such that $H \subseteq B$.

Infinite Dimensional Case

Let $K$ be a field.

Let $E$ be a vector space over $K$.

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

There exists a basis $B$ for $E$ such that $H \subseteq B$.

Linearly Independent Set is Basis iff of Same Cardinality as Dimension

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