Let F Rn rightarrow Rm be a linear transformation Show that

Let F : R^n rightarrow R^m be a linear transformation. Show that the following assertions are equivalent: F is one - to - one. Ker (F) = {0}.

Solution

Definition of one-to-one function is

If f:XY be a function.

f is one-to-one if and only if for every yY there is at most one xX such that f(x)=y;

In the problem if F is one-to-one then for every element of R^m, there is atmost one element of Rⁿ

It means after mapping every element, there is nothing left without mapping.

\'ker f\' is a set of all of the elements that were lost in the process of a mapping

If F is one-to-one there is nothing lost in the process of mapping. So

ker(F) = {0}

The two assertions

F is one-to-one and ker(f) = {0} are equivalent.