# 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.