Definition:Dimension of Vector Space

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 $n \in \N_{\ge 0}$.

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.