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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.
A transformation between these two vector spaces that also preserves the structure of vector spaces is a homomorphism.
A homomorphism that is also invertible is an isomorphism.
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]) ].