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.
Linearly Independent Set is Contained in some Basis
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.