Prove that m is a matroid

Reference no: EM131085718

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).

(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).

Reference no: EM131085718

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