Algebraic Techniques in Combinatorics - Yufei Zhao

MOP 2007 Black GroupAlgebraic Techniques in CombinatoricsYufei ZhaoAlgebraic Techniques in CombinatoricsJune 26, 2007 Yufei algebraUseful facts in linear algebra Any set ofn+ 1 vectors in ann-dimensional vector space is linearly dependent. That is, we canfind scalarsa1,..,an+1, not all zero, such thata1v1+ +an+1vn+1= 0. Almost all linear algebra results (especially the ones related to rank) are true over any field. Forinstance, the fieldF2is often useful for working with parity or incidence. However, note that wemight not be able to use eigenvalues. Suppose thatAis anm nmatrix, then the subspace spanned by its columns has the samedimension as the subspace spanned by its rows. This common dimension number is called therank ofA. In particular, we have rankA min(m,n). Rank-nullity theorem: rankA+ nullityA=n. rank(AB) min(rankA,rankB) rank(A+B) rankA+ rankB Ais invertible if and only ifm=n= rankA. IfAis a square matrix, then it is invertible if andonly if there does not exist a nonzero vectorvsuch thatAv= 0.

MOP 2007 Black Group Algebraic Techniques in Combinatorics Yufei Zhao Problem: (St. Petersburg) Students in a school go for ice cream in groups of at least two.

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  Technique, Algebraic, Combinatorics, Algebraic techniques in combinatorics

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