Basis extension
Task number: 2541
Prove that if \(V\) is a subspace of a finitely generated space \(W\) then there exist bases \(X\) and \(Y\), resp., of \(V\) and \(W\), resp., such that \(X \subseteq Y\).
Resolution
Choose any basis \(X\) of \(V\) and any auxiliary basis \(Y'\) of \(W\).
By the exchange theorem , (\(X\) is independent and \(Y'\) generates \(W\)) we find \(Y\) such that \(X\subseteq Y\), \(Y\) generates \(W\) and \(|Y|=|Y'|\). From the last two conditions follows that \(Y\) is the desired basis of \(W\).
