## Artin-Whaples Theorem

There seems to be another version of Artin-Whaples Theorem, called the Artin-Whaples Approximation theorem.

The theorem stated here is Artin-Whaples Theorem for central simple algebras.

Artin-Whaples Theorem: Let $A$ be a central simple algebra over a field $F$. Let $a_1,\dots,a_n\in A$ be linearly independent over $F$ and let $b_1,\dots,b_n$ be any elements in $A$. Then there exists $a_i',a_i''\in A$ for $i=1,\dots,m$ such that the $F$-linear map $f:A\to A$ defined by $f(x)=\sum_{r=1}^m a_r'xa_r''$ satisfies $f(a_j)=b_j$ for all $j=1,\dots,n$.

Very nice and useful theorem.