Dimension of a Vector Space

Last updated Nov 1, 2022

Let $V$ be a Finite-Dimensional Vector Space and let $S$ be a Vector Space Basis for $V$. Then $\dim V = |S|$. If $V$ is an Infinite-Dimensional Vector Space, we denote $\dim V = \infty$.

1. This is Well-Defined for all Vector Spaces because
   1. If there is a finite Vector Space Basis, then All Finite Bases for a Vector Space are the same Size.
   2. If there isn’t, then the dimension is $\infty$.
2. We don’t go into Set Cardinality here, for that see Cardinal Dimension of a Vector Space.