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JAIPUR NATIONAL UNIVERSITY, JAIPUR School of Distance Education & Learning Internal Assignment No. 1 Bachelor of Science (B. Sc) Paper Code: MT - 301 Paper Title: Algebra Last date of submission: Max. Marks: 15 Note: Question No. 1 is of short answer type and is compulsory for all the students. It carries 5 Marks. (Word limits 50-100) Q. 1. Answer all the questions: 1) What is the fundamental difference between a function and a relation? 2) Show that the set G = {1,w,w,2} of cubic roots of unity is an ableian group with respect to multiplication composition. 3) Find the identity element of Z if operation *, defined by a*b = a + b + 1 4) Let S= {1,2,3}, then symmetric set s3 of all permutation of degree 3 on s. 5) Consider the multiplicative group G = {1,-1, i,- i } and H= {-l,1 } be subgroup of G, then write all possible cosets of H. Note: Answer any two questions. Each question carries 5 marks (Word limits 500) Q.2 Show that the set Q+ of all positive rational numbers form a group under, the composition * on Qt, defined by a*b = ab v a,b £ Q. Q.3 A finite I ntegral domain is field. Q.4 Let S be an ideal of a ring R and T. be an ideal of R containing S then Q.5 Prove that if two vectors are linearly dependent, one of them is a scalar multiple of the other. JAIPUR NATIONAL UNIVERSITY, JAIPUR School of Distance Education & Learning Internal Assignment No. 2 Bachelor of Science (B. Sc) Paper Code: MT - 301 Paper Title: Algebra Last date of submission: Max. Marks: 15 Note: Question No. 1 is of short answer type and is compulsory for all the students. It carries 5 Marks. (Word limits 50-100) Q. 1. Answer all the questions: 1) Write the Descartes rule of signs. 2) Define the commutative rings and intregal domain. 3) Define the ring z and its properties. 4) Define the vector space over a field. 5) Define the group auto morphism and inner automorphism. Note: Answer any two questions. Each question carries 5 marks (Word limits 500) Q.2 Show that the set Q (√2) = { a+ b√2 : a ,b £ d y is a vector space over d w r to the compositions. (a+b√2) + (c+d√2) = (a+c) + (b+d)√2 (a +b√2) = Q.3 a+ Show that T : R4 b √2, £ d R4 defined by ƒ (x,y,z,t) = { x + y, x - y ,0, 0 } is a linear transformation. Find its rank and nullity. Q.4 Find the eigen values, eigen vectors and eigen spaces of A= ( )