Page 91 - DMTH502_LINEAR_ALGEBRA

Richa Nandra, Lovely Professional University Unit 5: Summary of Row-Equivalence

Unit 5: Summary of Row-Equivalence Notes

CONTENTS
Objectives
Introduction

5.1 Matrices and Elementary Row Operations
5.2 Row-reduced Echelon Matrices
5.3 Summary of Row-Equivalence

5.4 Summary
5.5 Keywords
5.6 Review Questions
5.7 Further Readings

Objectives

After studying this unit, you will be able to:

 Understand the technique of row operations on matrices of m × n type.
 Know that if B is a matrix obtained from row operations of A then B and A are called row
equivalent.

 Understand how to obtain a row reduced echelon matrix.

Introduction

In solving a system of simultaneous equations the method of row operations on m × n matrix
helps in finding the solution.
The idea of row space of a matrix helps in finding the subspace of the row space.

5.1 Matrices and Elementary Row Operations

Suppose F is a field. We consider the problem of finding x-scalars, x , x , ... x which satisfy the
1 2 n
conditions

A x A x 2 A x 3  A x n y 1
11 1
12
1n
13
A x A x A x  A x y
21 1 22 2 23 3 2n n 2 … (1)
    
A x A x A x  A x y
m 1 1 m 2 2 m 3 3 mn n n
where y ,...y and A , l < i < m, i  j  n are given elements of F. We shall now abbreviate the
1 m ij
system of equations (1) by

LOVELY PROFESSIONAL UNIVERSITY 85