# Definition:Dimension (Linear Algebra)

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## 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 $V$.

## 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