Definition:Dimension (Linear Algebra)/Vector Space

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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