Vector space/Basis/Exchange lemma/Fact/Proof
Proof
We show first that the new family is a generating system. Because of
and , we can express the vector as
Let be given. Then we can write
To show the linear independence, we may assume to simplify the notation. Let
be a representation of . Then
From the linear independence of the original family we deduce . Because of , we get . Therefore and hence for all .