Definition:Dimension (Linear Algebra)

Definition

Module

Let $R$ be a ring with unity.

Let $G$ be a free $R$-module which has a basis of $n$ elements.

Then $G$ is said to have a dimension of $n$ or to be $n$-dimensional.

The dimension of a free $R$-module $G$ is denoted $\map {\dim_R} G$, or just $\map \dim G$.

Vector Space

Let $K$ be a division ring.

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

Definition 1

The dimension of $V$ is the number of vectors in a basis for $V$.

Definition 2

The dimension of $V$ is the maximum cardinality of a linearly independent subset of $V$.

Work In Progress In particular: Infinite dimensional case You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by completing it. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{WIP}} from the code.

Sources

• 1992: George F. Simmons: Calculus Gems ... (previous) ... (next): Chapter $\text {B}.26$: Extensions of the Complex Number System. Algebras, Quaternions, and Lagrange's Four Squares Theorem