2013

(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

- Under what condition, two systems of linear equations over the same field are said to be equivalent?
- Write the standard basis for the vector space.
- Define null space of a linear transformation.
- Let be a linear map given by. What will be the kernel of ?

2. Answer the following questions:

- Find the dimension of the quotient space, where is the subspace of spanned by and. 2
- If and are the vectors ofandthen prove that 3

is linearly dependent.

- Show that the subset

of is a subspace of .

- Prove that the intersection of any two subspaces of a vector space is also a subspace of the vector space. 3
- Let be defined by

Find the matrix of T w.r.t. the standard bases of and respectively. 3

- If be the field of real numbers, then prove that the vectors (a, b) and (c, d) in are linearly dependent if and only if . 4

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

- Define affine space. Let be a subspace of a vector spaceand be fixed. Prove that

is an affine space.

- Let T be a linear transformation from V into W. then prove that T is non-singular if and only if T carries each linearly independent subset of V onto a linearly independent subset of W.
- Show that the mappingdefined as is a linear transformation from to. Find the rank, null space and nullity of T.
- If and are finite-dimensional subspaces of a vector space , then prove that is finite-dimensional and

GROUP – B

(Number Theory)

(Marks: 40)

4. Answer the following questions: 1x4=4

- Write the well-ordering principle (WOP) of positive integers.
- If and there exists such that, then write the value of.
- Define Euler’s function.
- Write a reduced set of residues mod 10.

5. Answer the following questions: 2x3=6

- Show that the difference between any integer and its cube is always divisible by 6.
- If g.c.d. then prove that

g.c.d.

- Under which situation, an arithmetic function is said to be a multiplicative function? Is the function defined as the sum of the divisors of, multiplicative?

6. Answer the following questions: 3x6=18

- Prove that if and , then

- Solve in integers:

- By the principle of mathematical induction, prove that is divisible by .
- Find the highest power of 5 which is contained in 500!.
- Is the system of linear congruence given below solvable? Give reasons for your answer:

- Find the value of the following:

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

- If , then prove that there exist integers and such that
- State Fermat’s little theorem. Using Fermat’s little theorem, find the remainder when is divided by 11.
- Prove that there are infinitely many primes.
- Prove that for

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