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27 Oct 2014: 0 3 1. . 3. Let A and B be nn matrices over a field F. ... Show that A can bediagonalised. 11. Let C be an nn matrix over C, and write C = A iB, where A and B are real nn matrices.

3 Dec 2014: If H(M) is free over Z, we can choose a basis ofH2n(M) and write (a,b) = aTAb for some symmetric matrix A with entries in Z. ... 14. Construct a 3 dimensional real vector bundle over S4 which does not have a nonvanishingsection.

22 Sep 2014: It is characterised up to homotopyequivalence over BO(2n) by these properties. ... Secondly, let hAut(θ) denote the grouplike topological monoid of self-homotopy. equivalences of B over BO(2n), i.e.

27 Oct 2014: Concretely we have. Objects Over b B: x E(b′) with u : b′ b.Maps From y with v : c′ c to x with u : b′ b :. c′v. - c. y ... have. Objects Over b B: x E(b′) with u : b′ b.Maps From y with v : c′ c to x with u : b′ b :. f b′F. - c′v. -

4 Dec 2014: 1. The square matrices A and B over the field F are congruent if B = PTAP for some invertible matrixP over F. ... Which, if any, over Q?) Try to get away with the minimum calculation.(.

27 Oct 2014: 0 3 1. . 3. Let A and B be nn matrices over a field F. ... Show that A can bediagonalised. 11. Let C be an nn matrix over C, and write C = A iB, where A and B are real nn matrices.

15 Jan 2014: PART II REPRESENTATION THEORYSHEET 4. Unless otherwise stated, all vector spaces are finite-dimensional over C. ...  : a,b,x Fp. of matrices over the finite field Fp (p prime).

26 Nov 2014: 1. The square matrices A and B over the field F are congruent if B = PTAP for some invertible matrixP over F. ... Which, if any, over Q?) Try to get away with the minimum calculation.(.

15 Jan 2014: b) Find an example of a representation of some finite group over some field of charac-teristic p, which is not completely reducible. ... Determine all finite groups which have a faithful 2-dimensional representation over R.

18 Sep 2014: The amalgamated free product of A and B. over α and β is the direct limit of this diagram. ... It is also the HNN-extension of Z over the triv-ial group:.