Which of the (nonlinear) transformations from to in Exercises 25 through 27 are invertible? Find the inverse if it exists.26. role="math" localid="1660365558437"
The inverse of the given transformation is
The given transformation is .
We can rewrite the transformation as .such that
Then the given transformation is invertible if and only if f is invertible.
(a) Suppose for some .Then and .So are zero or nonzero together.
Now and . So . If are nonzero, then and hence from (1) .If are zero, then .Thus and . Sofis one-one.
(b) Let . Then there exists such that . Therefore . Hence fis onto.
So,fis invertible and therefore the given transformation is invertible. Now Therefore inverse of the given transformation is
94% of StudySmarter users get better grades.Sign up for free