Linear Mapping

A linear mapping is a homomorphism between two vector spaces.

Description

Consider two vector spaces: V in Rⁿ 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 Rⁿ and a codomain of R^m.

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]) ].

CategoryRicottone