BSc IT Sem 5 Algebra L Question Paper PDF 2026 - Mumbai University | munotes

Questions asked in this paper

1. Q1 (a) Answer any and prove the Cayley Hamilton theorem 1 V bea finite dimensional inner product space over R and 7: V V be (8) inear transformation. Prove that the following statements are equivalent
   • (p) T is
   • (b) Answer any TWO
   • i. For real vector sgace V and a subspace W of V define quotient space V/W. Let V be a finite dimensional real vector space and W be a of V Show that dimV/W = dimV —dimW 6
   • ii. be a finite dimensional inner product space over R. If 7 : V V is an isometry, then show that there exists unique zp V and an orthogonal linear transformation T : V V such that = T(z) + 25, 6
   • iii. Find an orthogonal transformation in which represents reflection with
   • iv. Show that a 2 x 2 orthogonal matrix A with detA 1 is a matrix of rotation ' 2. (a) Answer any ONE
   • i. Define algebraic multiplicity and geometric multiplicity of an eigen value A. : of a real matrix A. Show that, if-A is ciagonalizable then (a) algebraic and geometric multiplicity of each eigen of A coincide (b) sum of geometric multiplicity of all the eigen values ai A is n 8
   • ii. Show that every real symmetric tmatrix is orthogonally diagonalizable. 8
   • (b) Answer any TWO
   • j. Show that every quadratic form = over R reduced to standard form by orthogonal change of variable X = PY, where X,Y are vectors of ii, Let A be upper triangular real matrix
   • (p) If all the main diagonal entries of A are distinct, then show that A is
   • (q) ifeach main diagonal entries of A is \ and A is diagonalizable then show
   • ii. Fin definite if and only if its eigen values are positive
2. Q3 (a) Answer any f order Prove that G contains a unique sub
   • i. G be a finite cyclic group of OF or d of group of two subgroup of G. Show that HK (
   • ii. Let G be a group and H and K KH be subgroup of G if and only if = — is a group homomorphism then define kernel of f that kernel of f is a subgroup of G Let G.G’ be groups and f : G be a onto homomorphism of groups
   • (p) If G is abelian then G' is
   • (q) G is cyclic and G = (a) then G’ is cyclic and G = iii, Let G = {5, 15, 25, 35} under multiplication of residue classes mod 40 composition table of G. State identity elernent of G and show that G is
   • iv. Give an example of a group G such that = 2 and = 5
   • (a) Let A, B ben real matrices. If A and are then prove that and BA are both orthogonal matrices
   • (b) Let . A linear T : — is defined by T(X) = AX (X being a vector sin Find kerT and ImT. Verify the fundamental. theorem of homumorphism of vector spaces incase of T
   • (d) associated with distinct eigen values of real symmetric
   • (e) Show thax a group G is abelian if and only if 7: G G define as f (z) = 2? is is a finite group an is a nonempty subset o th
   • (f) Ii G fi group and H bset of G then prove tha of G if and only if for any H, abe H

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