[BA 5th Sem Question Papers, Dibrugarh University, 2013, Mathematics, Major, Linear Algebra and Number Theory]
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 of
and
then prove that 3
is linearly dependent.
- Show that the subset
of 
is a subspace 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
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 space
and
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 mapping
defined 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:
Also Read: Dibrugarh University Question Papers
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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