Chapter 5 - Linear transformations and rank

Abstract

A linear transformation T is a mapping of a vector space Rn into a vector space Rm .The dimension of the range space R(T) is called the rank of the transformation and the dimension of the null space N(T) is called the nullity of T. The chapter also highlights the elementary matrices with the help of several examples. A matrix obtained from a unit matrix, by subjecting it to any of the elementary transformation is called an elementary matrix and it can be easily verified that every elementary matrix is nonsingular and each of the elementary row (column) transformation of a matrix may be considered as the result of the premultiplication (postmultiplication) with the corresponding elementary matrix. Analogous elementary transformations can be performed on the columns of a matrix by post multiplying the matrix by the corresponding elementary matrices obtained by the desired transformations on the columns of a unit matrix. Because elementary matrices are nonsingular, it is obvious that the product of a matrix with an elementary matrix does not change the rank of the matrix.