Definition: Let be a vector space. A linear transformation is left (right) invertible if ( ), for some linear transformation
Problem: Give an example of a vector space and a linear transformation such that is left but not right invertible.
Definition: Let be a vector space. A linear transformation is left (right) invertible if ( ), for some linear transformation
Problem: Give an example of a vector space and a linear transformation such that is left but not right invertible.
There are no such transformations between finite dimensional vector spaces : because then if is left invertible, then is injective, and hence surjective, and hence has both a left and a right inverse.
Consider the vector space consisting of sequences of elements in some field. Let be the "push" operator mapping . Then is clearly an injection, hence left invertible, but clearly is not a surjection, hence is not right invertible.