# Definition:Dimension (Linear Algebra)/Vector Space

## Contents

## Definition

Let $K$ be a division ring.

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

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

From Bases of Finitely Generated Vector Space have Equal Cardinality, all bases of $V$ have the same number of elements.

The **dimension** of a finite-dimensional $K$-vector space $V$ is denoted $\dim_K \left({V}\right)$, or just $\dim \left({V}\right)$.

Alternatively, the **dimension of $V$** can be defined as the maximum cardinality of a linearly independent subset of $V$.

The equivalence of these definitions follows from Size of Linearly Independent Subset is at Most Size of Finite Generator.

### Dimension of Vector

Informally, an element of an $n$-dimensional vector space is often referred to as an **$n$-dimensional vector**.

It must be understood that this is no more than a convenient shorthand. It is not the *vector* which possesses the dimensionality, but the *space in which it is embedded*.

## Finite Dimensional Vector Space

Let $V$ be a vector space which is $n$-dimensional for some $n \in \N_{>0}$.

Then $V$ is **finite dimensional**.

## Also see

## Sources

- 1964: Iain T. Adamson:
*Introduction to Field Theory*... (previous) ... (next): $\S 1.4$ - 1965: Seth Warner:
*Modern Algebra*... (previous) ... (next): $\S 27$ - 1969: C.R.J. Clapham:
*Introduction to Abstract Algebra*... (previous) ... (next): $\S 7.34$