## Sunday, December 30, 2018

2014
(November)
MATHEMATICS
(Major)
Course: 502
(Linear Algebra and Number Theory)
Full Marks: 80
Pass Marks: 32
Time: 3 hours
The figures in the margin indicate full marks for the questions
GROUP – A
(Linear Algebra)
(Marks: 40)

1. Answer the following questions: 1x4=4
1. Define subspace of a vector space.
2. Let be a vector space over same field . Show that 1. Define kernel of a linear mapping.
2. Find the matrix of the linear operator defined by with the usual basis.
1. Determine whether and are linearly dependent. 2
2. Let be a vector space over and . Show that is a subspace of iff
1. ;
2. . 3
1. Find, for what value of , the vector in is a linear combination of the vectors and . 3
2. Let be the subspace of generated by the vectors , , . Find a basis of .
3. Prove that a vector space of is the direct sum of its subspaces and , iff
1. 2. 1. Examine whether the following mapping are linear or not: 4
1. defined by 1. defined by 3. Answer any three of the following questions: 6x3=18
1. Show that any two bases of a finite-dimensional vector space have same number of elements.
2. Let be a linear transformation.
Show that
dim = rank of + nullity of 1. If is a finite-dimensional vector space of is a subspace of , then show that
dim = dim - dim .
1. Prove that the rows ranks and the column rank of an matrix are equal.

GROUP – B
(Number Theory)
(Marks: 40)
4. Answer the following questions: 1x4=4
1. If then show that 1. If denotes the largest integer , then find the value of 1. State Fermat’s little theorem.
2. What is the value of , if is prime?
5. Answer the following questions: 2x3=6
1. Prove that every non-empty subset of contains a least element.
2. If is a prime and then prove that or , where .
3. For any prime , who that .
6. Answer the following questions: 3x6=18
1. Prove that , if and only if 1. Find the remainder when is divided by 11.
2. Solve .
3. Solve the system 1. Prove that .
2. Prove that there are infinitely many primes of the form .
7. Answer any three of the following: 4x3=12
1. State and prove the division algorithm.
2. Prove that every positive integer can be expressed as a product of primes is unique apart from the order of factors.
3. Prove that 1. State and prove Chinese remainder theorem.

***