## 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. 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.

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