(10B1NMA731)
EXERCISE-1
APPLIED LINEAR ALGEBRA
Groups and Fields
1. Define the following terms(a) Binary operation
(b) Group
(c) Abelian group
2. Let G be the set of all polynomials of degree less than or equal to 3. Is (G, +) with usual
meaning of +, a group? Is it abelian?
3. Let S be the set of all m n matrices with elements as real numbers. Is (S, +) with usual meaning
of +, a group? Is it abelian?
4. Let S be the set of all n  n matrices with binary operations as the usual
multiplication (denoted by .). Is (S, .) a group? If no, what restrictions would you like to make so
that it becomes a group?
5. Let S be the set of all real valued functions defined over [0, 1]. Let the binary operation + on S
be defined by  f  g ( x)  f ( x)  g ( x), x  [0,1] ,  f , g  S . Is (S, +) a group? Is it abelian?
6. Let S =  - {-1}, i.e. the set of all real numbers except -1. Let  be the binary operation on S
defined by a  b  a  b  ab a, b  S with usual meanings to a  b and ab . Is (S,  ) a
group? Is it abelian?
7. In a group (G,  ) show that
(a) Identity element is unique.
(b)  element a there is a unique inverse a 1 .
(c) (a 1 ) 1  a
(d) (a  b) 1  b 1  a 1
(e) a  b  a  c  b  c (left cancellation law); a  b  c  b  a  c (right cancellation law)
(f) The equations a  x  b and x  a  b have unique solutions for x in G with given a and b
in G.
8. (a) Show that a group having only three elements is always abelian. (Note: This is true for any
group of order less than or equal to 5).
(b) Let G be the set of all solutions of z n  1 where z is a complex number and n is a nonnegative integer. Show that (G, .) is an abelian group under the usual multiplication.
9. Let G = {0, 1, 2,…, n-1} be the set of first n non-negative integers. Let  n be the binary
operation the addition (modulo n). Prove that (G,  n ) is an abelian group of order n. Prepare the
tables for n = 1,2,3.
10. Let G = {1, 2, 3, …, p-1} where p is a prime number. Also let . be the binary composition called
multiplication (modulo p). Show that (G, .) is an abelian group. Write tables for p = 2, 3, 5.
11. Show that the set of all 2  2 matrices with each entry equal to x , where x is a non-zero real
number, is an abelian group of singular matrices with matrix multiplication as binary operation.
12. Define the terms: (a) Field (b) Subfield.
13. Are the following sets fields with the usual operations addition(+) and multiplication(.)?
(a) Q , the set of all rational numbers
(b) R , the set of all real numbers,
(c) C , the set of all complex numbers.
Which one of them are subfields of others? Is ( Z ,,) a field, where Z is the set of integers?
14. Let Z n  {0,1,2,..., n  1} , where n is a positive integer. Define the binary operations  n (addition
modulo n) and  n (multiplication modulo n) on Z n . Verify that (Z n ,  n ,  n ) is a field for n = 5,
but not for n = 4. (Note: It is a field if n is a prime number).
15. Let Q[ 2] { a  b 2 | a, b Q} . Show that it is a field. (Note: Also, It can be shown that
Q[ p] with p as a prime number is field. Is Z[ 2] { a  b 2 | a, b Z} also a field? Give
reasons.
This is an assignment: Submit complete solutions of all questions
for evaluation on or before 28 August. It will be evaluated for your
Internal marks.