Systems of distinct representatives and linear algebra

Abstract

So me purposes of thi s pa per are: (1) To ta ke se riously the term , " term ra nk. " (2) T o ma ke a n iss ue of not " rea rra nging rows a nd co lu mn s" by not " a rran ging" th e m in th e firs t place. (3) To pro mote the nu me rica l use of C ra mer 's rul e. (4) To ill us tra te that the re le va nce of " numbe r of s te ps" to " a mount of wo rk " de pen d s on t he a mou nt of wo rk in a step. (5) To ca ll a tt e nti on to the com puta tional as pec t of SDR's, a n as pe ct wher e th e subjec t di ffe rs fro m be in g an insta nce of fa milia r li nea r alge bra. (6) To desc rib e a n SDR in s ta nce of a th eory on e xtre mal co m bi nato rics tha t uses lin ea r alge b ra in ve ry dif fe rent ways tha n does to tall y un imod ul ar t heo ry. (The preceding pape r, O ptimum Branc hin gs, de sc rib es a nothe r in sta nc e of tha t theory.)