# Vector Spaces The number of vectors in a *finite* [[basis]] of a [[vector space]] $V$ is the *dimension* of $V$. It can be shown that all *finite* bases have the same number of elements, so this definition is consistent. If no finite basis exists, then the space is *infinite dimensional*. The dimension of a vector space $V$ is often denoted $\text{dim}(V)$. # Manifolds A [[manifold]] $M$ of dimension $n$ locally "looks like" $\mathbb{R}^n$ ([[Real Coordinate Space|R^n]]. See the page on [[Manifold|manifolds]] for a more accurate definition of a manifold. # Matrices The dimension of a square [[Matrix|matrix]] is the number of columns or rows it has. # Representations The dimension of a [[group representation]] is the dimension of its representation [[vector space]], or in case of matrix representations, the dimension of the matrices used. # Lie Groups A [[Lie group|Lie group's]] dimension is given by the number of free parameters required to parameterize the group.The dimension of a Lie group is also the dimension of its manifold. See above. # Lie Algebras The dimension of the [[Lie Algebra|Lie algebra]] associated with a [[Lie group]] is the same as the dimension of the group.