Linear Algebra

A-9same role with respect to function composition that the number 0 plays in realnumber addition or that 1 plays in multiplication.In line with that analogy, we define a left inverse of a map f: X → Y to bea function g: range(f) → X such that g ◦ f is the identity map on X. A rightinverse of f is a h: Y → X such that f ◦ h is the identity.A map that is both a left and right inverse of f is called simply an inverse.An inverse, if one exists, is unique because if both g 1 and g 2 are inverses of fthen g 1 (x) = g 1 ◦ (f ◦ g 2 ) (x) = (g 1 ◦ f) ◦ g 2 (x) = g 2 (x) (the middle equalitycomes from the associativity of function composition), so we often call it “the”inverse, written f −1 . For instance, the inverse of the function f: R → R givenby f(x) = 2x − 3 is the function f −1 : R → R given by f −1 (x) = (x + 3)/2.The superscript ‘f −1 ’ notation for function inverse can be confusing sinceit clashes with 1/f(x). But it fits into a larger scheme. Functions that havethe same codomain as domain can be iterated, so that where f: X → X, we canconsider the composition of f with itself: f ◦ f, and f ◦ f ◦ f, etc. We write f ◦ f asf 2 and f ◦ f ◦ f as f 3 , etc. Note that the familiar exponent rules for real numbershold: f i ◦ f j = f i+j and (f i ) j = f i·j . Then where f is invertible, writing f −1 forthe inverse and f −2 for the inverse of f 2 , etc., gives that these familiar exponentrules continue to hold, once we define f 0 to be the identity map.If the codomain Y equals the range of f then we say that the function is onto.A function has a right inverse if and only if it is onto (this is not hard to check).If no two arguments share an image, if x 1 ≠ x 2 implies that f(x 1 ) ≠ f(x 2 ), thenthe function is one-to-one. A function has a left inverse if and only if it isone-to-one (this is also not hard to check).By the prior paragraph, a map has an inverse if and only if it is both ontoand one-to-one. Such a function is a correspondence. It associates one andonly one element of the domain with each element of the range. Because acomposition of one-to-one maps is one-to-one, and a composition of onto mapsis onto, a composition of correspondences is a correspondence.We sometimes want to shrink the domain of a function. For instance, we maytake the function f: R → R given by f(x) = x 2 and, in order to have an inverse,limit input arguments to nonnegative reals ˆf: R + → R. Then ˆf is a differentfunction than f; we call it the restriction of f to the smaller domain.Relations For some familiar operations we most naturally interpret them asfunctions: addition maps (5, 3) to 8. But what of ‘