Linear Transformations of Equations

Abstract

By a linear transformation of equation (A) we mean the process of passing from (A) to a new equation

$$ BAx = By $$

((10.1))

by means of a linear operator B which acts from F into a new Banach space G. It could be that the equation

$$ BAx = z $$

((BA))

is easier to solve than the original one. However, in dealing with such transformations we must proceed with a certain amount of caution. If the operator B is not defined everywhere on F, then the solutions of equation (A) corresponding to right-hand sides y ∉ υ(B) are not in the set of all solutions of equation (BA).