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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=
( )