# Definition:Dimension of Vector Space

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## Definition

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 $V$.

## 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**.

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

## 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*.

## Also see

- Size of Linearly Independent Subset is at Most Size of Finite Generator, proving equivalence of the definitions.

- Bases of Vector Space have Equal Cardinality: all bases of $V$ have the same number of elements.