### Inverse

##### 1. Inverse transformation

A transformation T: XY is invertible if there is a transformation S: YX satisfying

ST = idX, and TS = idY.

In other words, we have

ST(x) = x for any xX, and TS(y) = y for any yY.

Intuitively, the first equality means that S reverses what T does. The first equality means that T reverses what S does. We call S the inverse of T and denote

S = T-1.

Clearly, T is also the inverse of S: T = S-1.

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 = Reflection in x-axis, we have T2 = TT = id (reflecting twice goes back to itself).

Therefore we have T = T-1. The rotation Rθ by angle θ, is clearly reversed by the rotation R by angle -θ. Therefore we have R = Rθ-1.

For a non-invertible linear transformation, let us consider the zero transformation O (from R2 to R2, for example), that takes any vector to the zero vector. Clearly for any other linear transformation X, we have XO = O and OX = O. Therefore we cannot find an inverse for O.

One general way of computing the inverse is based on the following: If S is the inverse of T, then

T(x) = yS(y) = x.

Exercise To find the inverse of the transformation T(x) = 3x - 2: RR, we solve 3x - 2 = y and get x = y/3 + 2/3. Therefore T-1(y) = y/3 + 2/3.

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 et: R → (0, ∞). The exponential function can be sensibly defined as the function satisfying f' = f and f(0) = 1.

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