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10 Oct 2013: In whichof these cases is U a vector space over R?(a) x1 > 0.(b) either x1 = 0 or x2 = 0.(c) x1 x2 = 0.(d) x1 x2 = 1.(e) x1 ... 2. Determine which of the following sets of sequences of real numbers (xn) form vector spaces over R.(a) xn is bounded.(b) xn

3 Nov 2013: 17. Let L/K be an extension, and x,y L transcendental over K. ... Show that x is algebraicover K(y) iff y is algebraic over K(x).

12 Sep 2013: ii) The Pontrjagin–Thom construction provides a map. αg : B Diff (W2ng , D. ... iv) This means that the fibration sequence. R+(Wg ) R+(Wg )//Diff (Wg, D ) B Diff (Wg, D )is pulled back from a fibration over the -constructionB Diff (Wg, D ). , which

29 Oct 2013: . 2. Let A and B be n n matrices over a field F. ... Show that the minimum polynomial of A,over the complex numbers, has real coefficients.

2 Dec 2013: Leta K. Show that Xp a is irreducible over K if and only if it is irreducible over K′. ... τ/i R).(iii) Let L = Q(ζp). Show that L has a unique subfield K which is quadratic over Q, and thatK = Q(.

14 Nov 2013: Prove it?). 7. Which of the following symmetric matrices are congruent to the identity matrix (a) over C, (b) over Rand (c) over Q? ... 4 44 5. ). 1. 8. Find the rank and signature of the following quadratic forms over R.

27 Nov 2013: 7. 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. ... 1. 11. For A an n m and B an m n matrix over the field F , let τA(B) denote trAB.

3 Nov 2013: 17. Let L/K be an extension, and x,y L transcendental over K. ... Show that x is algebraicover K(y) iff y is algebraic over K(x).

1 Nov 2013: . 2. Let A and B be n n matrices over a field F. ... Show that the minimum polynomial of A,over the complex numbers, has real coefficients.

14 Nov 2013: Prove it?). 7. Which of the following symmetric matrices are congruent to the identity matrix (a) over C, (b) over Rand (c) over Q? ... 4 44 5. ). 1. 8. Find the rank and signature of the following quadratic forms over R.