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