if $X$ is independent and $E(M) - X$ is coindependent, show that $X$ is a basis and $E(M) - X$ is a cobasis.

Question

here is the question I am trying to solve:

In a matroid $M,$ if $X$ is independent and $E(M) - X$ is coindependent, show that $X$ is a basis and $E(M) - X$ is a cobasis.

I know how to prove that a set is a basis in a matroid $M$ iff its complement is a basis in the dual of M (using the rank function), however I do not know to prove the question I am trying to prove. Any help will be greatly appreciated.

Edit:

I have a lemma in Oxley that shows that $E(M) - X$ is independent in $M^*$ but how can we show that it is a maximal independent set?

Answer 1

If $X$ is independent, there exists a basis $B$ containing it, so that $X\subseteq B$, and so $E\setminus B\subseteq E\setminus X$, but if a basis is inside an independent set, it is because they are equal, because bases are maximal independent sets.