BSc Mathematics SEM V 2018 19 2018-19 Maths II Algebra Old 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 be a linear transformation. Prove that the following statements are
   • p. orthogonal
   • q. = for all X V
   • ii. State and prove the first isomorphism theorem for vector space. 8
   • (b) Answer any TWO
   • i. Let V be a vector space of finite dimension and W be of V Prove that dim V/W V— dim W
   • ii. Let W be an n dimensional inner product space and let W dimensional subspace of V. Let be a unit vector orthogonal to W Show that V + V defined by = x — is an orthogonal linear transformation such that T(w)=w, 6
   • iii. State the Cayley-Hamilton theorem. Using the theorem find
   • iv. Let a : R? > R? be defined by = + by +dy+ 6
   • f), where a,b,c,d,e, f R. Show that a is an isometry if and only if
2. Q2 (a) Answer any ONE
   • i. Show that every n real matrix with n eigen values is similar to an
   • ii. Let A ben xn real symmetric matrix. Show that the following state ments are equivalent
   • (p) (AX, X) > 0 for zero X
   • (q) Each eigen values of A is positive
   • (b) Answer any TWO
   • i. Define orthogonally diagonalizable matrix. Show that is orthogo- 6
   • ii. Show that a quadratic form Q[X] can be reduced to standard form Ay? by orthogonal change of variable X = PY, X = Y = and orthogonal matrix 6
   • iii. Show that characteristic roots of real symmetric matrix are real. 6
   • iv. Let be a non-zero real matrix such that A* for some k Show that characteristic polynomial of A is X” 6
3. Q3 (a) Answer any ONE
   • i. Define a cyclic group. Show that subgroup of a cyclic group is cyclic. Give an example to show that the converse is 8
   • ii. State and prove the 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 subgroup of group G if and only if HK = KA 6
   • iii. Let G = List all the subgroups of G and also list the generators of
   • iv. Show that the groups (Q,+) and (Q — are not isomorphic. 4, Answer any THREE 6
   • (a) Let V be a vector space and W be asubspace of V. Show for x,y V if and only EV
   • (b) If A is a 3 X 3 orthogonal matrix such that det(A) = 1. Show that lis an eigen value of A
   • (c) Let A be a diagonalizable matrix. Show that f(A) is also diagonalizable where f(x) is a polynomial over R
   • (d) Show that A = is diagonalizable but B= |0 1 0] is not
   • (e) Show that is a group where aob=ab/7 5
   • (f) Prove that f : GL,(R) defined by f(A) = det homomorphism. Show that f is onto but not one-one

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