AIS on Classical Groups and Introduction to K-theory 5-26 December at IISER Pune Assignment sheet 2 Exercise 1. Let V be a finite dimensional vector space over a field F: Prove that DimF (V ) = ∗ DimF (V ): Exercise 2. Let V be a vector space over F and let S ⊂ V: Show that ?L (?R (S)) ⊇ S and ?R (?L (S)) ⊇ S: Exercise 3. If B is non-degenerate on V and S is any subset of V show that ?L (?R (S)) =?R (?L (S)) is the subspace spanned by S: Exercise 4. Show that if B is a reflexive bilinear form on V and V = U ⊕ W; with U ? W: Show that rad(V ) = rad(U) ? rad(W ): 3 Exercise 5. Show that the inner product of vectors in R is a bilinear function and prove that it is non-degenerate. Exercise 6. Let C[a; b] denote the set of all continuous real valued functions on [a; b]: Show that R b the function α(f; g) = a f(x)g(x)dx is a non-degenerate bilinear function on C[a; b]: Exercise 7. Give an example of a degenerate and a non-degenerate bilinear form (not identically 2 2 zero) on C ⊕ C : Exercise 8. Let U and V be finite dimensional vector spaces and w be a bilinear form on U ⊕ V: If y0 2 V and if a function y is defined on U by y(x) = w(x; y0); then y is a linear functional on U: Is it true that if w is non-degenerate, then every linear functional on U can be obtained this way (by a suitable choice of y0?) Exercise 9. Let Pn be the set of all polynomials over C of degree less than or equal to n−1 together with the polynomial identically zero. Suppose that for each p(x); q(x) 2 Pn the function w : Pn ⊕ Pn ! C is defined by: R 1 (a) w(p; q) = 0 p(x)q(x)dx: (b) w(p; q) = p(1) + q(1): (c) w(p; q) = p(1)q(1): (d) w(p; q) = p(1)(dq=dx)x=1: Exercise 10. In which of the above cases is the bilinear form non-degenerate? 1 1 1 Exercise 11. Let X = Y = C ([a; b]): Then the operator defined by: B : C ([a; b]) × C ([a; b]) ! R R b 0 0 1 B(u; v) = x=a(uv + u v )dx is a bilinear form. (Here C ([a; b]) =) Exercise 12. Show that the function B(X; Y ) = trace(XY ) is a bilinear form on Mn(F ): 1 Exercise 13. Show that the determinant of matrices of order 2; regarded as a function of rows of 2 the matrix is a bilinear form on F : Determine the signature of the bilinear form when F = R: Exercise 14. Let V = M2(R): Show that the form B(A; B) = det(A + B) − det(A) − det(B) is a symmetric and bilinear form. Compute the matrix of this form with respect to the standard basis and determine its signature. Exercise 15. Let h ; i be a bilinear form on a real vector space V: Show that there exists a symmetric form ( ; ) and a skew symmetric form [ ; ] on V such that h ; i = ( ; ) + [ ; ]: p 3 Exercise 16. If K = Q( 2) and B : K × K ! Q is the trace form, B(x; y) = tryK=Q(xy), then find a matrix for B with respect to some basis of K over Q and find whether B is a non-degenerate symmetric bilinear form. Exercise 17. Repeat the above exercise for the trace forms for following algebras: (a) F [X]=(X2) over F (b) F (t)[X]=(X2 − t) over F , where F is a field of characteristic 2 and t is an indeterminate over F ..