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| 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. |
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| A homomorphism that is also [[LinearAlgebra/Invertibility|invertible]] is an '''isomorphism'''. | A homomorphism that also is [[Analysis/Injectivity|1-to-1]] and '''onto''' is an '''isomorphism'''. All isomorphisms are [[LinearAlgebra/Invertibility|invertible]]. |
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| If ''V'' and ''W'' are finite and [[LinearAlgebra/Basis|bases]] are defined for both, then all homomorphisms can be expressed with [[LinearAlgebra/MatrixMultiplication|matrix multiplication]]. | 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]) ]''. |
Linear Mapping
A linear mapping is a homomorphism between two vector spaces.
Contents
Description
Consider two vector spaces: V in Rn and W in Rm. A transformation T between these vector spaces is notated as T: V -> W. This transformation is said to have a domain of Rn and a codomain of Rm.
The result of transforming a vector u by T is called the 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 preserves the structure of vector spaces is a homomorphism.
A homomorphism that also is 1-to-1 and onto is an isomorphism. All isomorphisms are invertible.
Relation to Bases
If V and W are finite and bases are defined for both, then the homomorphism can be expressed with matrix multiplication. That matrix can be constructed by passing each standard basis in Rn through the transformation. In other words, if V is in R2, then the matrix is constructed as [ T([1 0]) T([0 1]) ].