math111_logo Transformation

4. Inverse

Recall a transformation T: XY is invertible if there is S: YX satisfying TS = idY and ST = idX. Because of the following characterization, an invertible transformation is also called a one-to-one correspondence.

For a transformation, invertible ⇔ onto and one-to-one.

Proof By this characterization, T: XY is onto means there is S: YX satisfying TS = idY. By this characterization, T: XY is one-to-one means there is S': YX satisfying S'T = idX. Thus

T: XY is onto and one-to-one ⇔ There are S, S': YX satisfying TS = idY and S'T = idX.

The proof is then complete by showing that S and S' must be equal:

TS = idY and S'T = idXS = idXS = S'TS = S'idY = S'.


[part 1] [part 2] [part 3] [part 4] [part 5] [part 6]