Vector Space Dimensions
Why
The number of vectors in any basis is the same.
Defining result
A vector space is finite-dimensional if it has a finite basis; otherwise it is it is infinite-dimensional.
Every basis of a finitely spanned vector space
has the same number of elements.
The dimension of a finite-dimensional vector space is the number of distinct vectors in any basis. If a vector space is finite-dimensional and every basis has $n$ distinct elements we call it a $n$-dimensional vector space.