Image (mathematics) in the context of Injective function

In mathematics, a family, or indexed family, is informally a collection of objects, each associated with an index from some index set. For example, a family of real numbers, indexed by the set of integers, is a collection of real numbers, where a given function selects one real number for each integer (possibly the same) as indexing.

More formally, an indexed family is a mathematical function together with its domain ${\displaystyle I}$ and image ${\displaystyle X}$ (that is, indexed families and mathematical functions are technically identical, just points of view are different). Often the elements of the set ${\displaystyle X}$ are referred to as making up the family. In this view, an indexed family is interpreted as a collection of indexed elements, instead of a function. The set ${\displaystyle I}$ is called the index set of the family, and ${\displaystyle X}$ is the indexed set.

In mathematics, given two groups, (G,∗) and (H, ·), a group homomorphism from (G,∗) to (H, ·) is a function h : G → H such that for all u and v in G it holds that

where the group operation on the left side of the equation is that of G and on the right side that of H.

In mathematics, a codomain or set of destination of a function is a set into which all of the outputs of the function is constrained to fall. It is the set Y in the notation f: X → Y. The term range is sometimes ambiguously used to refer to either the codomain or the image of a function.

A codomain is part of a function f if f is defined as a triple (X, Y, G) where X is called the domain of f, Y its codomain, and G its graph. The set of all elements of the form f(x), where x ranges over the elements of the domain X, is called the image of f. The image of a function is a subset of its codomain so it might not coincide with it. Namely, a function that is not surjective has elements y in its codomain for which the equation f(x) = y does not have a solution.

In linear algebra and functional analysis, a projection is a linear transformation ${\displaystyle P}$ from a vector space to itself (an endomorphism) such that ${\displaystyle P\circ P=P}$. That is, whenever ${\displaystyle P}$ is applied twice to any vector, it gives the same result as if it were applied once (i.e. ${\displaystyle P}$ is idempotent). It leaves its image unchanged. This definition of "projection" formalizes and generalizes the idea of graphical projection. One can also consider the effect of a projection on a geometrical object by examining the effect of the projection on points in the object.

In mathematics, the range of a function may refer either to the codomain of the function, or the image of the function. In some cases the codomain and the image of a function are the same set; such a function is called surjective or onto. For any non-surjective function ${\displaystyle f:X\to Y,}$ the codomain ${\displaystyle Y}$ and the image ${\displaystyle {\tilde {Y}}}$ are different; however, a new function can be defined with the original function's image as its codomain, ${\displaystyle {\tilde {f}}:X\to {\tilde {Y}}}$ where ${\displaystyle {\tilde {f}}(x)=f(x).}$ This new function is surjective.

In projective geometry, a collineation is a one-to-one and onto map (a bijection) from one projective space to another, or from a projective space to itself, such that the images of collinear points are themselves collinear. A collineation is thus an isomorphism between projective spaces, or an automorphism from a projective space to itself. Some authors restrict the definition of collineation to the case where it is an automorphism. The set of all collineations of a space to itself form a group, called the collineation group.

In mathematics, a projection is a mapping from a set to itself—or an endomorphism of a mathematical structure—that is idempotent, that is, equals its composition with itself. The image of a point or a subset ⁠${\displaystyle S}$⁠ under a projection is called the projection of ⁠${\displaystyle S}$⁠.

An everyday example of a projection is the casting of shadows onto a plane (sheet of paper): the projection of a point is its shadow on the sheet of paper, and the projection (shadow) of a point on the sheet of paper is that point itself (idempotency). The shadow of a three-dimensional sphere is a disk. Originally, the notion of projection was introduced in Euclidean geometry to denote the projection of the three-dimensional Euclidean space onto a plane in it, like the shadow example. The two main projections of this kind are:

In mathematics, particularly linear algebra, an orthonormal basis for an inner product space ${\displaystyle V}$ with finite dimension is a basis for ${\displaystyle V}$ whose vectors are orthonormal, that is, they are all unit vectors and orthogonal to each other. For example, the standard basis for a Euclidean space ${\displaystyle \mathbb {R} ^{n}}$ is an orthonormal basis, where the relevant inner product is the dot product of vectors. The image of the standard basis under a rotation or reflection (or any orthogonal transformation) is also orthonormal, and every orthonormal basis for ${\displaystyle \mathbb {R} ^{n}}$ arises in this fashion.An orthonormal basis can be derived from an orthogonal basis via normalization.The choice of an origin and an orthonormal basis forms a coordinate frame known as an orthonormal frame.

For a general inner product space ${\displaystyle V,}$ an orthonormal basis can be used to define normalized orthogonal coordinates on ${\displaystyle V.}$ Under these coordinates, the inner product becomes a dot product of vectors. Thus the presence of an orthonormal basis reduces the study of a finite-dimensional inner product space to the study of ${\displaystyle \mathbb {R} ^{n}}$ under the dot product. Every finite-dimensional inner product space has an orthonormal basis, which may be obtained from an arbitrary basis using the Gram–Schmidt process.

The term surjective and the related terms injective and bijective were introduced by Nicolas Bourbaki, a group of mainly French 20th-century mathematicians who, under this pseudonym, wrote a series of books presenting an exposition of modern advanced mathematics, beginning in 1935. The French word sur means over or above, and relates to the fact that the image of the domain of a surjective function completely covers the function's codomain.

In mathematics, a twisted cubic is a smooth, rational curve C of degree three in projective 3-space P. It is a fundamental example of a skew curve. It is essentially unique, up to projective transformation (the twisted cubic, therefore). In algebraic geometry, the twisted cubic is a simple example of a projective variety that is not linear or a hypersurface, in fact not a complete intersection. It is the three-dimensional case of the rational normal curve, and is the image of a Veronese map of degree three on the projective line.

In algebra, the kernel of a homomorphism is the relation describing how elements in the domain of the homomorphism become related in the image. A homomorphism is a function that preserves the underlying algebraic structure in the domain to its image.

When the algebraic structures involved have an underlying group structure, the kernel is taken to be the preimage of the group's identity element in the image, that is, it consists of the elements of the domain mapping to the image's identity. For example, the map that sends every integer to its parity (that is, 0 if the number is even, 1 if the number is odd) would be a homomorphism to the integers modulo 2, and its respective kernel would be the even integers which all have 0 as its parity. The kernel of a homomorphism of group-like structures will only contain the identity if and only if the homomorphism is injective, that is if the inverse image of every element consists of a single element. This means that the kernel can be viewed as a measure of the degree to which the homomorphism fails to be injective.

In mathematics, the disjoint union (or discriminated union) ${\displaystyle A\sqcup B}$ of the sets A and B is the set formed from the elements of A and B labelled (indexed) with the name of the set from which they come. So, an element belonging to both A and B appears twice in the disjoint union, with two different labels.

A disjoint union of an indexed family of sets ${\displaystyle (A_{i}:i\in I)}$ is a set ${\displaystyle A,}$ often denoted by ${\textstyle \bigsqcup _{i\in I}A_{i},}$ with an injection of each ${\displaystyle A_{i}}$ into ${\displaystyle A,}$ such that the images of these injections form a partition of ${\displaystyle A}$ (that is, each element of ${\displaystyle A}$ belongs to exactly one of these images). A disjoint union of a family of pairwise disjoint sets is their union.

In set theory, the axiom schema of replacement is a schema of axioms in Zermelo–Fraenkel set theory (ZF) that asserts that the image of any set under any definable mapping is also a set. It is necessary for the construction of certain infinite sets in ZF.

The axiom schema is motivated by the idea that whether a class is a set depends only on the cardinality of the class, not on the rank of its elements. Thus, if one class is "small enough" to be a set, and there is a surjection from that class to a second class, the axiom states that the second class is also a set. However, because ZFC only speaks of sets, not proper classes, the schema is stated only for definable surjections, which are identified with their defining formulas.

In machine learning, kernel machines are a class of algorithms for pattern analysis, whose best known member is the support-vector machine (SVM). These methods involve using linear classifiers to solve nonlinear problems. The general task of pattern analysis is to find and study general types of relations (for example clusters, rankings, principal components, correlations, classifications) in datasets. For many algorithms that solve these tasks, the data in raw representation have to be explicitly transformed into feature vector representations via a user-specified feature map: in contrast, kernel methods require only a user-specified kernel, i.e., a similarity function over all pairs of data points computed using inner products. The feature map in kernel machines is infinite dimensional but only requires a finite dimensional matrix from user-input according to the representer theorem. Kernel machines are slow to compute for datasets larger than a couple of thousand examples without parallel processing.

Kernel methods owe their name to the use of kernel functions, which enable them to operate in a high-dimensional, implicit feature space without ever computing the coordinates of the data in that space, but rather by simply computing the inner products between the images of all pairs of data in the feature space. This operation is often computationally cheaper than the explicit computation of the coordinates. This approach is called the "kernel trick". Kernel functions have been introduced for sequence data, graphs, text, images, as well as vectors.