Results concerning Generators and Bases of Vector Spaces

Theorem

Generator of Vector Space Contains Basis

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

Every generator for $G$:

$(1): \quad$ has at least $n$ elements;

$(2): \quad$ contains a basis for $G$;

$(3): \quad$ is a basis for $G$ iff it contains exactly $n$ elements.

Cardinality of Linearly Independent Set No Greater than Dimension

Every linearly independent subset of $G$:

$(1): \quad$ has at most $n$ elements

$(2): \quad$ is contained in a basis for $G$

$(3): \quad$ is a basis for $G$ iff it contains exactly $n$ elements.

Sources

• Seth Warner: Modern Algebra (1965): $\S 27$: Theorem $27.14$