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A function is injective (one-to-one) if each possible element of the codomain is mapped to by at most one argument. Equivalently, a function is injective if it maps distinct arguments to distinct images. An injective function is an injection. [1] The formal definition is the following.
In mathematics, an injective function (also known as injection, or one-to-one function [1]) is a function f that maps distinct elements of its domain to distinct elements of its codomain; that is, x 1 ≠ x 2 implies f(x 1) ≠ f(x 2) (equivalently by contraposition, f(x 1) = f(x 2) implies x 1 = x 2).
The function f : R → R defined by f(x) = x 3 − 3x is surjective, because the pre-image of any real number y is the solution set of the cubic polynomial equation x 3 − 3x − y = 0, and every cubic polynomial with real coefficients has at least one real root. However, this function is not injective (and hence not bijective), since, for ...
The projection of a onto b is often written as or a ∥b. The vector component or vector resolute of a perpendicular to b , sometimes also called the vector rejection of a from b (denoted oproj b a {\displaystyle \operatorname {oproj} _{\mathbf {b} }\mathbf {a} } or a ⊥ b ), [ 1 ] is the orthogonal projection of a onto the plane (or ...
Let X be an affine space over a field k, and V be its associated vector space. An affine transformation is a bijection f from X onto itself that is an affine map; this means that a linear map g from V to V is well defined by the equation () = (); here, as usual, the subtraction of two points denotes the free vector from the second point to the first one, and "well-defined" means that ...
Recognizing that their Numerical Recipes books were increasingly valued more for their explanatory text than for their code examples, the authors significantly expanded the scope of the book, and significantly rewrote a large part of the text. They continued to include code, still printed in the book, now in C++, for every method discussed. [5]
Figure 1. 2D majorization example. For , , we have if and only if is in the convex hull of all vectors obtained by permuting the coordinates of .This is equivalent to saying that = for some doubly stochastic matrix. [2]:
Octave (aka GNU Octave) is an alternative to MATLAB. Originally conceived in 1988 by John W. Eaton as a companion software for an undergraduate textbook, Eaton later opted to modify it into a more flexible tool. Development began in 1992 and the alpha version was released in 1993. Subsequently, version 1.0 was released a year after that in 1994.