Definition:Dimension (Linear Algebra)

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Let $R$ be a ring with unity.

Let $G$ be a free $R$-module which has a basis of $n$ elements.

Then $G$ is said to have a dimension of $n$ or to be $n$-dimensional.

The dimension of a free $R$-module $G$ is denoted $\map {\dim_R} G$, or just $\map \dim G$.

Vector Space

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 $V$.