Inverse InverseA transformation T: X → Y is invertible if there is a transformation S: Y → X satisfying
ST = idX, and TS = idY.
In other words, we have
ST(x) = x for any x ∈ X, and TS(y) = y for any y ∈ Y.
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-θ by angle -θ. Therefore we have R-θ = 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) = y ⇔ S(y) = x.
Exercise To find the inverse of the transformation T(x) = 3x - 2: R → R, 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.