M.Sc.(Semester II) (CBCS) Examination, 2017
MATHEMATICS
FUNDAMENTAL IN MATHEMATICS
Day Date: Friday, 28-04-2017 Max. Marks: 70
Time: 10.30 AM to 01.00 PM
N.B. Attempt five questions.
Q no. 1 Q. no. 2 are compulsory.
Attempt any three questions from Q. no. 3 to Q. no. 7.
Figures to the right indicate full marks.
Q.1 Choose the correct Answer: 07
Vector space is defined over a
Group Ring Field None of these
A subset of Linearly independent vectors is
Linearly dependent
Linearly independent
May or may not be linear dependent
None of these
If v V is the identity map, then Rank T
0 1 2 dim V
If no. of equations equal to no. of unknowns then
solution exists for Non-Homogeneous system of equation.
Unique Infinite No solution Finite
If A is a square matrix of order 4 with rank 2 B is a square
matrix of order 4 whose rank is 4 then the rank of AB is
1 2 3 4
If dim V=n then the number of vectors in a basis of V is
Less than n Equal to n
Greater than n None of these
If T → is a linear map V is finite dimensional,
then Rank
dim V nullity dim V nullity
nullity dim V nullity
State whether true or false. 07
Any system of linear equations has at least one solution.
The rank of n x n matrix is at most n.
An elementary matrix is always square.
The zero vector space has no basis.
The dimension of fn is n.
Page 2 of 2
A vector space may have more than one zero vector.
The empty set is linearly dependent.
Q.2
Find the rank of matrix A

03
Show that the inverse of matrix is unique. 04
Show that R2→R2 by defined a2) a2) T is linear
transformation.
04
Define Basis Dimension of vector space. 03
Q.3 Prove that any Intersection of subspace of vector space is
subspace.
07
Show that the following system of equation is consistent find
its solution.
07
Q.4 Prove that If a vector space V is generated by a finite set S then
some subset of S is basis for V Hence V has a finite basis.
08
Determine whether or not the following vectors form a basis of
R3.


06
Q.5 State prove Rank-Nullity theorem. 08
Prove that: If V be vector space is Linearly
dependent then is linearly dependent.
06
Q.6 Show that R2 R2 defined by a2) a1) is both
one-one and onto.
07
Explain the properties of Determinants with example. 07
Q.7 Define Elementary matrix explain elementary matrix
operations.
06
Define T R2 R3 by
a2) a1, 3a2, 2a1 4a2]
U R2 R3 by U a2) a1 a2, 2a1 3a1 2a2]
Consider be the ordered basis for R2 R3 respectively