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21 May 2005: Find the dual basis for R3. 4. Let V be a 4-dimensional vector space over R, and let {ξ1,ξ2,ξ3,ξ4} be the basis of V dual to the ... Verify that (DS) = SD and (SD) = DS. 16. For A an n m and B an m n matrix over the field F, let τA(B) denote

21 May 2005: 5. Let A and B be n n matrices over a field F. ... 6. Let C be an n n matrix over C, and write C = A iB, where A and B are real n n matrices.

26 Sep 2005: where g = det(gij), dx = dx1 dxn, dy = dy1 dyn, and the symbol ̂ over. ... We integrate this identity over SM against the Liouville measure, using the flowinvariance of the measure and (27):.

4 Apr 2005: Clearly A is itself an admissible subcategory of A. The Grothendieck group K0(B) of B is the abelian group with generators [M]where M runs over all the objects of ... Because tensor products commute with finite directsums, part (b) follows. Note that the

12 Apr 2005: Summing over all vertices of U we obtain the result. The next step in the chain of reasoning is the following innocent-looking statement. ... the following idea. If f is a function of three variables x, y and z that range over sets X,.