if $X \subseteq Y $ are flats s.t. $r(Y) = r(X)-1$ then $\exists$ hyperplane $H$ s.t. $Y = H \cap X$.

Question

Let $X$ and $Y$ be flats of a matroid $M$ such that $Y \subseteq X$ and $r(Y) = r(X)-1$. How can i prove that $M$ has a hyperplane $H$ such that $Y = H \cap X$ ?

Answer 1

You can do this just as you would for a representable matroid. Take a basis for $Y$, then add an element from $X$ to get a basis for $X$. Extend this to a basis for the entire ground set, then delete the single element of $X$ to get a basis for your hyperplane.