linear maps - Definition. Was ist linear maps
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Was (wer) ist linear maps - definition

MAPPING THAT PRESERVES THE OPERATIONS OF ADDITION AND SCALAR MULTIPLICATION
Linear operator; Linear mapping; Linear transformations; Linear operators; Linear transform; Linear maps; Linear isomorphism; Linear isomorphic; Linear Transformation; Linear Transformations; Linear Operator; Homogeneous linear transformation; User:The Uber Ninja/X3; Linear transformation; Bijective linear map; Nonlinear operator; Linear Schrödinger Operator; Vector space homomorphism; Vector space isomorphism; Linear extension of a function; Linear extension (linear algebra); Extend by linearity; Linear endomorphism

linear map         
  • The function f:\R^2 \to \R^2 with f(x, y) = (2x, y) is a linear map. This function scales the x component of a vector by the factor 2.
  • The function f(x, y) = (2x, y) is additive: It doesn't matter whether vectors are first added and then mapped or whether they are mapped and finally added: f(\mathbf a + \mathbf b) = f(\mathbf a) + f(\mathbf b)
  • The function f(x, y) = (2x, y) is homogeneous: It doesn't matter whether a vector is first scaled and then mapped or first mapped and then scaled: f(\lambda \mathbf a) = \lambda f(\mathbf a)
<mathematics> (Or "linear transformation") A function from a vector space to a vector space which respects the additive and multiplicative structures of the two: that is, for any two vectors, u, v, in the source vector space and any scalar, k, in the field over which it is a vector space, a linear map f satisfies f(u+kv) = f(u) + kf(v). (1996-09-30)
Linear map         
  • The function f:\R^2 \to \R^2 with f(x, y) = (2x, y) is a linear map. This function scales the x component of a vector by the factor 2.
  • The function f(x, y) = (2x, y) is additive: It doesn't matter whether vectors are first added and then mapped or whether they are mapped and finally added: f(\mathbf a + \mathbf b) = f(\mathbf a) + f(\mathbf b)
  • The function f(x, y) = (2x, y) is homogeneous: It doesn't matter whether a vector is first scaled and then mapped or first mapped and then scaled: f(\lambda \mathbf a) = \lambda f(\mathbf a)
In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that preserves the operations of vector addition and scalar multiplication. The same names and the same definition are also used for the more general case of modules over a ring; see Module homomorphism.
linear transformation         
  • The function f:\R^2 \to \R^2 with f(x, y) = (2x, y) is a linear map. This function scales the x component of a vector by the factor 2.
  • The function f(x, y) = (2x, y) is additive: It doesn't matter whether vectors are first added and then mapped or whether they are mapped and finally added: f(\mathbf a + \mathbf b) = f(\mathbf a) + f(\mathbf b)
  • The function f(x, y) = (2x, y) is homogeneous: It doesn't matter whether a vector is first scaled and then mapped or first mapped and then scaled: f(\lambda \mathbf a) = \lambda f(\mathbf a)

Wikipedia

Linear map

In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V W {\displaystyle V\to W} between two vector spaces that preserves the operations of vector addition and scalar multiplication. The same names and the same definition are also used for the more general case of modules over a ring; see Module homomorphism.

If a linear map is a bijection then it is called a linear isomorphism. In the case where V = W {\displaystyle V=W} , a linear map is called a linear endomorphism. Sometimes the term linear operator refers to this case, but the term "linear operator" can have different meanings for different conventions: for example, it can be used to emphasize that V {\displaystyle V} and W {\displaystyle W} are real vector spaces (not necessarily with V = W {\displaystyle V=W} ), or it can be used to emphasize that V {\displaystyle V} is a function space, which is a common convention in functional analysis. Sometimes the term linear function has the same meaning as linear map, while in analysis it does not.

A linear map from V to W always maps the origin of V to the origin of W. Moreover, it maps linear subspaces in V onto linear subspaces in W (possibly of a lower dimension); for example, it maps a plane through the origin in V to either a plane through the origin in W, a line through the origin in W, or just the origin in W. Linear maps can often be represented as matrices, and simple examples include rotation and reflection linear transformations.

In the language of category theory, linear maps are the morphisms of vector spaces.