specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function)...

In linear algebra, the transpose of a linear map between two vector spaces, defined over the same field, is an induced map between the dual spaces of...

In mathematics, linear maps form an important class of "simple" functions which preserve the algebraic structure of linear spaces and are often used as...

related areas of mathematics, a continuous linear operator or continuous linear mapping is a continuous linear transformation between topological vector...

conjugate of s . {\displaystyle s.} Antilinear maps stand in contrast to linear maps, which are additive maps that are homogeneous rather than conjugate homogeneous...

associated vector spaces over the field k. A map f : X → Z is an affine map if there exists a linear map mf : V → W such that mf (x − y) = f (x) − f (y)...

+ ⋯ + a n x n = b , {\displaystyle a_{1}x_{1}+\cdots +a_{n}x_{n}=b,} linear maps such as ( x 1 , … , x n ) ↦ a 1 x 1 + ⋯ + a n x n , {\displaystyle (x_{1}...

polynomial. An example of a linear function is the function defined by f ( x ) = ( a x , b x ) {\displaystyle f(x)=(ax,bx)} that maps the real line to a line...

piecewise linear or segmented function is a real-valued function of a real variable, whose graph is composed of straight-line segments. A piecewise linear function...

of paper. The term map may be used to distinguish some special types of functions, such as homomorphisms. For example, a linear map is a homomorphism of...

Piecewise linear function Linear approximation Linear interpolation Discontinuous linear map Linear least squares "The term linear function means a linear form...

codomain. A linear map has this property if and only if it is identically 0. {\displaystyle 0.} Consequently, in functional analysis, when a linear operator...

through a linear map V ⊗ W → Z {\displaystyle V\otimes W\to Z} (see the section below titled 'Universal property'), i.e. the bilinear map is associated...

natural linear map F → V ⊗ V'; in the language of linear maps, it assigns to a scalar c the linear map c⋅idV. Sometimes this is called coevaluation map, and...

In mathematics, a linear fractional transformation is, roughly speaking, an invertible transformation of the form z ↦ a z + b c z + d . {\displaystyle...

mathematics, a bilinear map is a function combining elements of two vector spaces to yield an element of a third vector space, and is linear in each of its arguments...

{\displaystyle C^{op}\to D.} A linear map f : V → W {\displaystyle f:V\to W\,} gives rise to a corresponding linear map f ¯ : V ¯ → W ¯ {\displaystyle...

Together with the preceding property, this implies that the transpose is a linear map from the space of m × n matrices to the space of the n × m matrices. (...

mathematics, the kernel of a linear map, also known as the null space or nullspace, is the part of the domain which is mapped to the zero vector of the co-domain;...

analysis, spaces of linear maps between two vector spaces can be endowed with a variety of topologies. Studying space of linear maps and these topologies...

the Banach–Steinhaus theorem) states that a set H {\displaystyle H} of linear maps between Banach spaces is equicontinuous if it is pointwise bounded; that...

space. For general index sets I, a linear map between two I-graded vector spaces f : V → W is called a graded linear map if it preserves the grading of homogeneous...

In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called vectors, can be added together and multiplied...

lesser of the number of rows and columns, and the rank. The rank of a linear map or operator Φ {\displaystyle \Phi } is defined as the dimension of its...

In mathematics, a linear form (also known as a linear functional, a one-form, or a covector) is a linear map from a vector space to its field of scalars...

with a group that acts by linear transformations of the space is called a linear representation of the group. A linear map that commutes with the action...

is linear as a map over the real numbers R {\displaystyle \mathbb {R} } . If X and Y are complex vector spaces then A may fail to be linear as a map over...

In linear algebra, a multilinear map is a function of several variables that is linear separately in each variable. More precisely, a multilinear map is...

this matrix is viewed as a linear map). More generally, the differential or pushforward refers to the derivative of a map between smooth manifolds and...

is defined as the set of all linear maps φ : V → F {\displaystyle \varphi :V\to F} (linear functionals). Since linear maps are vector space homomorphisms...