Definition:Dimension of Vector Space

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, 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.

Also see

 * Bases of Vector Space have Equal Cardinality