Page 65 - DMTH502_LINEAR_ALGEBRA

Unit 1: Vector Space over Fields

Notes
19. In ( )F x let V be the set of all polynomials of degree less than n. Using the natural
n
operations for polynomials of addition and multiplication by a F, show that V is a
n
vector space over F(x).
1.6 Summary

 The concept of set is fundamental in all branches of mathematics. A set according to the
German mathematician George Cantor, is a collection of definite well-defined objects of perception
or thought. By a well defined collection we mean that there exists a rule with the help of
which it is possible to tell whether a given object belongs or does not belong to the given
collection.
 Let A and B be two sets. The union of A and B is the set of all elements which are in set A
or in set B. We denote the union of A and B by A B, which is usually read as “A union B”.
On the other hand, the intersection of A and B is the set of all elements which are both in
A and B. We denote the intersection of A and B by A B, which is usually read as “A
intersection B”.
 The properties of natural numbers were developed in a logical manner for the first time
by the Italian mathematician G. Peano, by starting from a minimum number of simple
postulates. These simple properties, are known as the Peano’s Postulates (Axioms).

 The system of rational numbers Q provides an extension of the system of integral Z, such
that (i) Q Z, (ii) addition and multiplication of two integers in Q have the same meanings
as they have in Z and (iii) the subtraction and division operations are defined for any two
numbers in Q, except for division by zero.

1.7 Keywords

Complex Number: The product set R × R consisting of the ordered pairs of real numbers.

Fields: A commutative ring with unity is called a field if its every non-zero element possesses a
multiplicative inverse.
Irrational Number: A real number which cannot be put in the form p/q where p and q are
integers.
Modulus of the Complex Number z: If z = (a, b) be any complex number, then the non-negative
real number (a b 2 ) .

Operator or Transformation of A: If the domain and co-domain of a function f are both the same
set say f : A A, then f is often called the operator.

Tabular form of the Set: Here the elements are separated by commas and are enclosed in
brackets { }

1.8 Review Questions

n
1. Let S be a set of all real numbers of the form (m 2 ) where m, n Q, a set of rational
number, prove that S is a multiplication or additive group, m, n not vanishing
simultaneously.

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