By Maak W.

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The dot product of a vector with a unit coordinate vector is the projection of the vector along that coordinate direction. A vector is decomposed into the sum of noninteracting components by projecting it along mutually orthogonal unit vectors. Within the general framework introduced in this chapter, signals of various kinds are seen to act like vectors. They can be added and subtracted. Norms are defined, so that the closeness of one signal to another, or the degree to which one signal approximates another, can be quantified.

Frequently, we write f : X → Y to say “f is a mapping from X to Y”. ” For each point x ∈ X, the corresponding point y ∈ Y is called the image of x, and is denoted y = f (x). If each point x in the domain has only one image point, then the mapping f is called a function. That is, a function is a single-valued mapping. In this text, when a function’s domain is a subset of the integers, X ⊂ ℤ, we will denote the image of n by f [n] rather than f (n). A function whose domain is a set of successive integers, for example, X = {1, 2, … , N}, is also called a sequence.

If a point y ∈ Y is the image of a point x ∈ X, we call x the preimage of y. An inverse f −1 can be defined using preimages. For a point y ∈ Y, f −1 (y) = {preimages of y in X} = {x ∈ X ∣ f (x) = y}. The inverse may or may not be a function, in that y may have more than one preimage. If every y in the range of f has a unique preimage, we say f is one-to-one or injective. If the range of f is identically Y, that is, if every y ∈ Y has a preimage in X, we say that f is onto or surjective. 9 Mappings f from points in a set X to points in a set Y.