Left inverse

A function g is a left inverse of f when g∘f is the identity on f’s domain. Such a function exists exactly when f is injective.

Function composition: An operation that combines functions by applying one function to the output of another. The equation g∘f = id defines a left inverse through composition.
Right inverse: A function h is a right inverse of f when f∘h is the identity on f’s codomain. Unlike a left inverse, a right inverse requires f to reach every element of its codomain.
Function (mathematics): A relation assigning exactly one output to each element of a domain. The left-inverse equation applies to functions with specified domains and codomains.
Left inverse of a matrix: A matrix L satisfying LA = I for a given matrix A. It is the linear-algebra version of undoing a map after applying it.
Injective function: A function that maps distinct inputs to distinct outputs. A left inverse can recover each input only if f never merges distinct inputs.
Bijective function: A function that is both injective and surjective. A function has both a left and right inverse precisely when it is bijective.
Domain of a function: The set of inputs on which a function is defined. The identity required by g∘f acts on f’s domain.
Full column rank: A matrix property stating that its columns are linearly independent. A matrix has a left inverse exactly when its columns are linearly independent.
Identity function: A function that maps every element of a set to itself. Composing f with its left inverse must return this function on f’s domain.
Inverse function: A function that reverses another function under composition on both sides. A two-sided inverse combines the left-inverse equation with the right-inverse equation.

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