Given a linear transformation T(x) in functional form, its transformation matrix can be constructed by applying T to each vector of the standard basis, then inserting the results into the columns of the new matrix.

A transformation matrix of dimension n×m operates on a column vector of dimension m×1 to produce a row vector of dimension 1×n.

1963 [McGraw-Hill], Lawrence P. Huelsman, Circuits, Matrices and Linear Vector Spaces, 2011, Dover, page 191,
We would like to make as many as possible of the elements of the transformation matrix equal zero.

1968 [McGraw-Hill], Granino A. Korn, Theresa M. Korn, Mathematical Handbook for Scientists and Engineers, 2000, Dover, page 414,
Refer to Sec. 14.8-6 for a procedure yielding transformation matrices T with the desired properties.

The 4x4 transformation matrices are conveniently used to convert various entities expressed in different coordinate systems into another.

Transformation matrix

Meaning

Noun