Given a set of [vectors](Linear%20Algebra%20and%20Matrix%20Theory%20(index).md#Vectors) $\{v_1,...,v_m\}$, if its [span](Linear%20span.md) is equal to a [vector space,](Vector%20spaces.md) $\mathcal{V},$ then $\mathcal{V}$ is a _finite dimensional vector space._ For a finite dimensional vector space we may also define $\mathrm{span}(v_1,...,v_m)$ where we say $\{v_1,...,v_m\}$ _spans_ the vector space. ^661403 %%mention here the basis are the vectors in the span%% # 0 dimensional vector space A vector space that [only contains the $0$ vector](Vector%20spaces.md#The%20zero%20vector%20space) $\mathbf{0},$ is also a _0 dimensional vector space_. ^39b11c For [such a vector space](Finite%20dimensional%20vector%20spaces.md#0%20dimensional%20vector%20space) the [basis](linear%20basis) is the [the empty set](The%20null%20set.md) $\{\},$ %%A 0 dimensional vector space is NOT an empty set.%% # Finite subspaces %%Recommended reading that describes vector space dimensionality for finite vectors will likely be the section on that in Halmos's text on vector spaces. Fill that in later.%% ## Invariant subspaces of finite vector spacs Every [real](Real%20vector%20spaces.md) [nonzero](Vector%20spaces.md#The%20zero%20vector%20space) [finite vector space](Finite%20dimensional%20vector%20spaces.md) has a $1$ or $2$ [dimensional](Vector%20space%20dimension.md) [invariant subspace.](Invariant%20subspaces.md) %%This statement is from Axler's book pg 91. This is kind of weird. complex vector spaces aren't like this, are they?%% # Examples of Finite Dimensional Vector Spaces ## The 3 Dimensional Euclidean space  It is also true that the vector space dimension is 3, corresponding to the three spatial dimensions in physical space. #MathematicalFoundations/Algebra/AbstractAlgebra/LinearAlgebra/VectorSpaces