Linear Algebra

A-9We sometimes instead use the notation x ↦−→ f16x 2 − 100, read ‘x maps underf to 16x 2 − 100’, or ‘16x 2 − 100 is the image of x’.Some maps, like x ↦→ sin(1/x), can be thought of as combinations of simplemaps, here, g(y) = sin(y) applied to the image of f(x) = 1/x. The compositionof g : Y → Z with f : X → Y , is the map sending x ∈ X to g( f(x) ) ∈ Z. It isdenoted g ◦ f : X → Z. This definition only makes sense if the range of f is asubset of the domain of g.Observe that the identity map id: Y → Y defined by id(y) = y has theproperty that for any f : X → Y , the composition id ◦ f is equal to f. So anidentity map plays the same role with respect to function composition thatthe number 0 plays in real number addition, or that the number 1 plays inmultiplication.In line with that analogy, define a left inverse of a map f : X → Y to be afunction g : range(f) → X such that g ◦ f is the identity map on X. Of course,a right inverse 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 — itdoesn’t mean 1/f(x). It is used because it fits into a larger scheme. Functionsthat have the same codomain as domain can be iterated, so that wheref : X → X, we can consider the composition of f with itself: f ◦f, and f ◦ f ◦f,etc. Naturally enough, we write f ◦ f as f 2 and f ◦ f ◦ f as f 3 , etc. Notethat the familiar exponent rules for real numbers obviously hold: f i ◦ f j = f i+jand (f i ) j = f i·j . The relationship with the prior paragraph is that, where f isinvertible, writing f −1 for the inverse and f −2 for the inverse of f 2 , etc., givesthat these familiar exponent rules continue to hold, once f 0 is defined to be theidentity 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 ),then the 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 and onlyone element of the domain with each element of the range (for example, finite