= Linear Mapping = A '''linear mapping''' is a homomorphism between two vector spaces. <> ---- == Description == Consider two vector spaces: ''V'' in ''R^n^'' and ''W'' in ''R^m^''. A '''transformation''' ''T'' between these vector spaces is notated as ''T: V -> W''. This transformation is said to have a '''domain''' of ''R^n^'' and a '''codomain''' of ''R^m^''. The result of transforming a vector ''u'' by ''T'' is called the [[Analysis/Functions|image]] of ''T(u)''. Conversely, the input vector that is transformed by ''T'' into ''u'' is called the '''pre-image''' of ''u''. A transformation maps all members of the space ''V'' into ''W'', but it does not necessarily span all of ''W''. The subspace that is spanned by ''T(V)'' is called the '''range''' of ''T'', or the image of ''T(V)''. The range is always a subset of the codomain. A transformation between these two vector spaces that also [[LinearAlgebra/Linearity|preserves the structure of vector spaces]] is a '''homomorphism'''. A homomorphism that also is [[Analysis/Injectivity|1-to-1]] and [[Analysis/Surjectivity|onto]] is an '''isomorphism'''. All isomorphisms are [[LinearAlgebra/Invertibility|invertible]]. === Relation to Bases === If ''V'' and ''W'' are finite and [[LinearAlgebra/Basis|bases]] are defined for both, then the homomorphism can be expressed with [[LinearAlgebra/MatrixMultiplication|matrix multiplication]]. That matrix can be constructed by passing each standard basis in ''R^n^'' through the transformation. In other words, if ''V'' is in ''R^2^'', then the matrix is constructed as ''[ T([1 0]) T([0 1]) ]''. ---- CategoryRicottone