All Bases are Same Size Theorem

If $S=\{x_1,\dots ,x_k\}$ and $S^{\prime }=\{y_1,\dots ,y_n\}$ are two finite bases of Vector Space $V$, then $n=k$.

This means that all bases are the same size. Allows for the Subspace Dimension function

Proof

Case 1: Consider $S$ as the spanning set Consider $S^{\prime }$ as the indep set. Then: $n\le k$ by The Dimension Bound Theorem.

Case 2: Consider $S^{\prime }$ as the spanning set Consider $S$ as the independent set This gives $k\le n$ again by The Dimension Bound Theorem Therefore, $n=k$ $\square$