MathDB

5

Part of 2009 IMC

Problems(2)

IMC 2009 Day 1 P5

Source:

7/15/2014
Let nn be a positive integer. An n-\emph{simplex} in Rn\mathbb{R}^n is given by n+1n+1 points P0,P1,⋯ ,PnP_0, P_1,\cdots , P_n, called its vertices, which do not all belong to the same hyperplane. For every nn-simplex S\mathcal{S} we denote by v(S)v(\mathcal{S}) the volume of S\mathcal{S}, and we write C(S)C(\mathcal{S}) for the center of the unique sphere containing all the vertices of S\mathcal{S}. Suppose that PP is a point inside an nn-simplex S\mathcal{S}. Let Si\mathcal{S}_i be the nn-simplex obtained from S\mathcal{S} by replacing its ithi^{\text{th}} vertex by PP. Prove that : ∑j=0nv(Sj)C(Sj)=v(S)C(S) \sum_{j=0}^{n}v(\mathcal{S}_j)C(\mathcal{S}_j)=v(\mathcal{S})C(\mathcal{S})
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IMC 2009 Day 2 P5

Source:

7/17/2014
Let M\mathbb{M} be the vector space of m×pm \times p real matrices. For a vector subspace S⊆MS\subseteq \mathbb{M}, denote by δ(S)\delta(S) the dimension of the vector space generated by all columns of all matrices in SS. Say that a vector subspace T⊆MT\subseteq \mathbb{M} is a \emph{covering matrix space} if ⋃A∈T,A≠0ker⁡A=Rp \bigcup_{A\in T, A\ne \mathbf{0}} \ker A =\mathbb{R}^p Such a TT is minimal if it doesn't contain a proper vector subspace S⊂TS\subset T such that SS is also a covering matrix space.
(a) (8 points) Let TT be a minimal covering matrix space and let n=dim⁡(T)n=\dim (T) Prove that δ(T)≤(n2) \delta(T)\le \dbinom{n}{2} (b) (2 points) Prove that for every integer nn we can find mm and pp, and a minimal covering matrix space TT as above such that dim⁡T=n\dim T=n and δ(T)=(n2)\delta(T)=\dbinom{n}{2}
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