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In this Note we show that the notion of a basis of a finite-dimensionalvector space could be introduced by an argument much weaker than Gauss'reduction method. Our aim is to give a short proof of a simply formulatedlemma, which in fact is equivalent to the theorem on frame extension, usingonly a simple notion of the kernel of a linear mapping, without any referenceto special results, and derive the notions of basis and dimension in a quiteintuitive and logically appropriate way, as well as obtain their basicproperties, including a lucid proof of Steinitz's theorem.

On the notion of a basis of a finite dimensional vector space