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20 Nov 2015: 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.(.

11 Feb 2015: 7. For a Serre fibration p : E B with connected fibre F over b0, and simply connectedbase, consider the maps of fibrations. ... Fi //. Ep //. p. B. {b0} // B B. and show using the naturality of the Serre spectral sequence that the compositions.

12 Feb 2015: i) Show that if B is finite over A, and C is finite over B, then C is finite over A. ... ii) Show that if B is integral over A, and C is integral over B, then C is integralover A.

9 Nov 2015: 14. Let C be an nn matrix over C, and write C = A iB, where A and B are real nn matrices. ... Deduce that if two n n real matrices P and Q are similar whenregarded as matrices over C, then they are similar as matrices over R.

26 Oct 2015: 3. Let V be a 4-dimensional vector space over R, and let {ξ1,ξ2,ξ3,ξ4} be the basis of V dual to the basis{x1,x2,x3,x4} for ... 9. Let A and B be nn matrices over a field F.

23 Nov 2015: 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.(.

19 Jan 2015: 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).

10 Nov 2015: 14. Let C be an nn matrix over C, and write C = A iB, where A and B are real nn matrices. ... Deduce that if two n n real matrices P and Q are similar whenregarded as matrices over C, then they are similar as matrices over R.

26 Oct 2015: 3. Let V be a 4-dimensional vector space over R, and let {ξ1,ξ2,ξ3,ξ4} be the basis of V dual to the basis{x1,x2,x3,x4} for ... 9. Let A and B be nn matrices over a field F.

27 Feb 2015: Nj=1 ‖γ(tj) γ(tj1)‖ where the sup is taken over all. finite partitions 0 = t0 < t1 <. < tN = 1. (i) Give an example for which (γ) =. If γ is continuously ... b) = inf (γ), wherethe inf is taken over all continuous γ: [0, 1] E with γ(0) = a,