A linear transformation is a function from one vector space to another that respects the underlying (linear) structure of each vector space. A linear transformation is also known as a linear operator or map. The two vector spaces must have the same underlying field. A transformation matrix is a matrix that represents a linear transformation in linear algebra. These have specific applications to the world of computer programming and machine learning. The matrix of a linear transformation is like a snapshot of a person; here are many pictures of a person, but only one person. Likewise, a given linear transformation can be represented by matrices with respect to many choices of bases for the domain and range. This also means that applying the transformation T to a vector is the same as multiplying by this matrix. Such a matrix can be found for any linear transformation. There is only one standard matrix for any given transformation and it is found by applying the matrix to each vector in the standard basis of the domain.

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