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 \(W\).

Difficulty level: Easy task (using definitions and simple reasoning)
Proving or derivation task
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