2015

(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. (a) Write when a system of linear equations is said to be consistent. 1

(b) Show that the additive inverse of any vector in a vector space is unique. 2

(c) Find the solution set for the following system of equations: 3

And

2. Answer any two of the following:

- Let and be two subspaces of a vector space over a field. Show that is also a subspace of. 4
- When is a set of vectors in a vector space said to be linearly independent? Examine whether the vectors and of are linearly independent over, where are non-zero real numbers. 1+3=4
- Show that the vectors, and from a basis for

4

3. Define subspace of a vector space with an example. If is a proper subspace of a finite dimensional vector space, then prove that is finite dimensional and. 2+3=5

4. Answer any three of the following:

- Define a line in a vector space. Prove that any two distinct points determine a unique line in any vector space. 1+2=3
- Let be a vector subspace of a vector space and be a fixed vector in. Prove that the set is an affine space of. 3
- Let be a subspace of a vector space over a field. Show that two cosets and are equal if and only if where. 3
- Let be the subspace of spanned by and. Find the basis for the quotient space. 3

5. Answer any two of the following:

- Let be the mapping defined by

Prove that is a linear mapping. Also find the basis and the dimension of the image of

2+3+1=6

- Define a linear operator on a vector space with an example. Let be a linear operator on given by . Find the matrix of with respect to the basis where and. 2+4=6
- Define isomorphism of vector spaces. Prove that the mapping

from to is an isomorphism. 2+4=6

GROUP – B

(Number Theory)

(Marks: 40)

6. (a) If and are relatively prime, then prove that and . 2

(b) Find the integers and such that. 2

(c) Use division algorithm to establish that the cube of any integer has one of the forms or. 3

7. (a) Find the number of divisors of 3 and 5 in between 500 and 1000. 3

(b) Find the highest power of 5 which is contained in 200!. 2

(c) If and are two integers such that, then show that or . 3

8. (a) Write a reduced set of residues modulo 9. 1

(b) Find the remainder when the following sum is divided by 15: 3

(c) Write the condition that is to be satisfied so that gives . 1

(d) Find the remainder when is divided by 7. 2

(e) Does there exist any solution for the linear congruence If so, find all of them. 1+4=5

(f) Solve: 3

9. (a) If and are positive integers with , then prove that 2

(b) Find the value. 2

(c) Show that ifis a prime number, then

2

(d) Define a multiplicative arithmetic function. Prove that the Mobius -function is multiplicative function. 1+3=4

Or

Prove that there are infinitely many primes. 4

***

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