whats the difference between reversible and 'one to one' transformation?

if by reversible you mean "invertible", then the same distinction that applies to functions apply. for reversible we need the transformation to be one to one and onto. so the distinction is just that. reversible transformation are one-to-one transformations with an additional property. that is, they are also onto (surjective)

see Linear Algebra

scroll down to the section that says, "Inverse". check the two links in the first bullet

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