A transformation ` T`:

` ST` =

In other words, we have

` ST`(

Intuitively, the first equality means that ` S` reverses what

` S` =

Clearly, ` T` is also the inverse of

More discussions on inverse transformations can be found here.

Example The transformation

`Capitalize`: {`a`, `b`, `c`, ..., `z`} → {`A`, `B`, `C`, ..., `Z`}

has

`de-Capitalize`: {`A`, `B`, `C`, ..., `Z`} → {`a`, `b`, `c`, ..., `z`}

as the inverse transformation. The transformation

`Name`: {+, -, >, <, =, 0, 1} → {plus, minus, bigger, smaller, equal, zero, one}

has

`Symbol`: {plus, minus, bigger, smaller, equal, zero, one} → {+, -, >, <, =, 0, 1}

as the inverse transformation.

For the transformations

`Capital City`: country → city,

`Which Country`: city → country,

we have

(`Which Country`)(`Capital City`) = `id`

(`Capital City`)(`Which Country`) ≠ `id`.

Thus the two transformations are not inverse to each other. In fact, neither transformation is invertible.

Example
For ` T` =

Therefore we have ` T` =

For a non-invertible linear transformation, let us consider the zero transformation ` O`
(from

One general way of computing the inverse is based on the following: If ` S` is the inverse of

` T`(

Exercise To find the inverse of the transformation
` T`(

We also recall that one way to introduce the log function `ln`: (0, ∞) → **R** is to
define it as the inverse of the exponential function `e ^{t}`:

You may also recall how the inverse trigonometric functions are defined.

[Extra: Criterion for invertibility of a transformation]