Definition:Dimension of Vector Space

Definition
Given a vector space $V$, 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 Linearly Independent Subset of Finitely Generated Vector Space.

Dimensions of a 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.