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

- Define subspace of a vector space.
- Let be a vector space over same field. Show that

- Define kernel of a linear mapping.
- Find the matrix of the linear operator defined by

with the usual basis.

2. Answer the following questions:

- Determine whether and are linearly dependent. 2
- Let be a vector space over and. Show that is a subspace of iff

- ;
- . 3

- Find, for what value of, the vector in is a linear combination of the vectors and. 3
- Let be the subspace of generated by the vectors, , . Find a basis of.
- Prove that a vector space of is the direct sum of its subspaces and , iff

- Examine whether the following mapping are linear or not: 4

- defined by

- defined by

3. Answer any three of the following questions: 6x3=18

- Show that any two bases of a finite-dimensional vector space have same number of elements.
- Let be a linear transformation.

Show that

dim = rank of + nullity of

- If is a finite-dimensional vector space of is a subspace of , then show that

dim = dim - dim .

- 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

- If then show that

- If denotes the largest integer , then find the value of

- State Fermat’s little theorem.
- What is the value of, if is prime?

5. Answer the following questions: 2x3=6

- Prove that every non-empty subset of contains a least element.
- If is a prime and then prove that or, where.
- For any prime, who that.

6. Answer the following questions: 3x6=18

- Prove that , if and only if

- Find the remainder when is divided by 11.
- Solve.
- Solve the system

- Prove that.
- Prove that there are infinitely many primes of the form.

7. Answer any three of the following: 4x3=12

- State and prove the division algorithm.
- Prove that every positive integer can be expressed as a product of primes is unique apart from the order of factors.
- Prove that

- State and prove Chinese remainder theorem.

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