Rota's basis conjecture

a friend posed this to me last night and left me to stew on it for an hour before telling me it was an open problem:

Let \(B_1 = \{b_{1,1}, \cdots , b_{1,n}\}, \cdots B_n\) be \(n\) mutually disjoint bases for some finite-dimensional vector space \(V \simeq \bigoplus_{i \leq n} F^i\) over a field \(F\). Consider a matrix having each \(B_i\) as its \(i^{\mathrm{th}}\) row. Do there exist permutations on each of the rows so that the columns of the resulting matrix are also all bases of \(V\), i.e. do there exist

\[\sigma_1, \cdots \sigma_n \in S_n : \{\{b_{1,\sigma_1(i)}\}_{i \leq n}, \{b_{2, \sigma_2(i)}\}_{i \leq n}, \cdots , \{b_{n, \sigma_n (i)}\}_{i \leq n}\}\]

so that the columns of this matrix are also all bases of \(V\)?

thoughts, anybody? the best i could get is iterating induction + pigonhole to get a sequence of nested \(n \times n, n-1 \times n-1, \cdots\) matrices whose first columns were linearly independent.

bagarang said:

Teu tumblr é mt foda, leq, tenho q me segurar p n sair dando rt em td hahaha

Nuss valeu vei KK,  pode ir se soltando e reblogando mais KKKK

Das Regal aus Salzgitter

Das Regal aus Salzgitter

Foto: F. Wehrmann

Die Amerikanerin Phillys Rose hat ein Buch über ein aberwitziges Abenteuer geschrieben: “The Shelf“, zu deutsch das Bücherregal.

Frau Rose, eine Vielleserin, hatte nach der Suche zu einem Buch eine Idee: Sie suchte sich willkürlich ein Regal aus der “New York Society Library” aus und las alle dort stehenden Bücher der Reihe nach durch – von den Signaturen LEQ bis…

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