Swati said:Let T:V->V be a linear operator on an n-dimensional vector space. Prove that exactly one of the following statements holds:
(i) the equation T(x)=b has a solution for all vectors b in V.
(ii) Nullity of T>0
Use the rank-nullity theorem again, to show that T is surjective.Swati said:how to proof if second statement is true then first statement is false.
Yes of course. I should have said: Use the rank-nullity theorem again, to show that T is NOT surjective.Deveno said:if the SECOND statement is true, T CANNOT be surjective:
by the rank-nullity theorem:
dim(V) = rank(T) + nullity(T).
if nullity(T) > 0, then rank(T) < dim(V), so that:
dim(im(T)) < dim(V).
thus there is some b in V not in im(T).
(i only posted this because Opalg's post answers the wrong question).
A linear transformation is a mathematical operation that maps one vector space to another in a linear manner. It is often represented by a matrix and can be used to describe various physical phenomena in science, such as the motion of particles or the flow of fluids.
The Fredholm Alternative Theorem is a fundamental theorem in linear algebra that provides a condition for the existence and uniqueness of solutions to linear equations. It states that either the equation has a unique solution or the associated homogeneous equation has a non-trivial solution.
In science, the Fredholm Alternative Theorem is used to analyze and solve linear equations that arise in various fields, such as physics, engineering, and economics. It is also used in optimization problems and in the study of differential equations.
The Fredholm Alternative Theorem is significant because it provides a powerful tool for solving linear equations and understanding the behavior of linear systems. It also has applications in many areas of science and engineering, making it an important concept for scientists to understand.
No, the Fredholm Alternative Theorem only applies to linear systems. However, there are other theorems and techniques that can be used to analyze and solve non-linear systems, such as the Banach Fixed Point Theorem and the Newton-Raphson method.