Let m s i be a matroid prove that mlowast is a matroid

Question

Math 121c: Topics in Geometric Combinatorics, Spring 2012 Problems-

Let M = (S, I) be a matroid.

(a) Prove that M^∗ is a matroid.

(b) Prove that the rank function r^∗ of M^∗ is given by r^∗(A) = |A| - r(M) + r(S\A), and conclude T_M∗ (x, y) = T_M(y, x).

(c) Show that if e ∈ E(M) is not a loop nor a coloop, then M/e and M\e are matroids.

(d) Suppose e ∈ S is not a loop nor coloop. Describe M^∗, M\e, M/e if

• M is U_r,n with 1 < r < n.
• M is a linear matroid (i.e. M is consists of the columns of a matrix with entries in some field).