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A linear transformation between topological vector spaces, for example normed spaces, may be continuous. If its domain and codomain are the same, it will then be a continuous linear operator. A linear operator on a normed linear space is continuous if and only if it is bounded, for example, when the domain is finite-dimensional. [18]
In linear algebra, linear transformations can be represented by matrices.If is a linear transformation mapping to and is a column vector with entries, then there exists an matrix , called the transformation matrix of , [1] such that: = Note that has rows and columns, whereas the transformation is from to .
Linear maps are mappings between vector spaces that preserve the vector-space structure. Given two vector spaces V and W over a field F, a linear map (also called, in some contexts, linear transformation or linear mapping) is a map: That is compatible with addition and scalar multiplication, that is
A linear isometry also necessarily preserves angles, therefore a linear isometry transformation is a conformal linear transformation. Examples. A linear map from to itself is an isometry (for the dot product) if and only if its matrix is unitary. [10] [11] [12] [13]
The linear combinations relating the first basis to the other extend to a linear transformation, called the change of basis. A vector represented by two different bases (purple and red arrows). In mathematics , an ordered basis of a vector space of finite dimension n allows representing uniquely any element of the vector space by a coordinate ...
A transformation A ↦ P −1 AP is called a similarity transformation or conjugation of the matrix A. In the general linear group , similarity is therefore the same as conjugacy , and similar matrices are also called conjugate ; however, in a given subgroup H of the general linear group, the notion of conjugacy may be more restrictive than ...
In contrast, a linear map is a function that is additive and homogeneous, where is called homogeneous if = (). An antilinear map f : V → W {\displaystyle f:V\to W} may be equivalently described in terms of the linear map f ¯ : V → W ¯ {\displaystyle {\overline {f}}:V\to {\overline {W}}} from V {\displaystyle V} to the complex conjugate ...
linear form A linear map from a vector space to its field of scalars [8] linear independence Property of being not linearly dependent. [9] linear map A function between vector space s which respects addition and scalar multiplication. linear transformation A linear map whose domain and codomain are equal; it is generally supposed to be invertible.