BSc Mathematics SEM V 2016 17 2016-17 Mathematics II Algebra Question Paper - Mumbai University | munotes

Questions asked in this paper

• (2) Figures to the right indicate marks for respective subquestions

1. Q1 (a) Answer any ONE
   • i. State and prove the first isomorphism theorem of vector space. 8
   • ii. Let V be a finite dimensional inner over R. If V isa map such that (i) f(0) = 0 Gi) = = V, then show that f is an orthogonal linear transformation 8
   • (b) Answer any TWO
   • i. Let W be a subspace of vector space V, define V/W. Show that. following operations are well defined on V/W = 6
   • ii. Define orthogonal linear transformation. For finite dimensional inner prod- uct space V, V > a linear transformation, prove that following statements are equivalent : 6
   • (p) T is orthogonal
   • (q) If is an orthonormal basis of V then is also an or thonormal basis of
   • iii. Show that a 2 x 2 orthogonal matrix A with det A = —1 is a matrix of 6
   • iv. Let V W = {A Tr(A) = 0} be a subspace of V. Find bases of W and V/W 6
2. Q2 (a) Answer any ONE
   • i. Show that ann real matrix A is diagonalizable if and only if R” has a basis consisting of eigen vectors of A 8
   • ii. Show that every real symmetric matrix of order n is orthogonally diagonal- 8
   • (b) Answer any TWO
   • i. Show that the characteristic roots of a real symmetric matrix are real. 6
   • ii. Let symmetric matrix. Prove that AX -Y = AY for every X,Y column vectors. Hence or otherwise prove that eigen vectors corresponding to distinct eigen values of a real symmetric matrix are 6
   • iii. Let A= Find a non-singular matrix P s.t. is a diagonal matrix and find Al 6
   • iv. Define positive definite real symmetric matrix. Show that, if is a positive definite real symmetric matrix then all the eigen values of A are 6
   • Q.P. Code :05713
3. Q3 (a) Answer any ONE
   • i. Let G be a cyclic group of order n generated by a. Prove that generates G if and only if ged (m,n) = 1 8
   • ii. State and prove Lagrange’s theorem. 8
   • (b) Answer any TWO
   • i. Show that an infinite cyclic group has only two generators. 6
   • ii. Prove that if H and K are subgroups of a group G then Hk is a subgroup of group G if and only if HK = KH 6
   • iii. For a group G, (ab)? = (ab)* = (ab)? = for all a,b G then show that G is abelian 6
   • iv. Let f :G— be a group homomorphism. Prove that : 6
   • p. f(e) =e, where are identity elements of G and G’ respectively
4. Q4 Answer any THREE
   • (a) Let A= [4 0]. Using the Cayley-Hamilton theorem, find A*+ — 5
   • (b) Let A be 3 x 3 orthogonal matrix such that det A = —1. Show that —1 is an eigen value of A 5
   • (c) Show that matrix A having n distinct eigen values is diagonalizable. 5
   • (d) Define rank and signature of the quadratic form the rank and sig- nature of Q[X] = 2y? — 2ry 5
   • (e) Show that the set {5, under multiplication modulo 40 is a group. 5
   • (f) Show that the group G = is isomorphic to the group C — non-zero complex numbers under multiplication 5

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