Transformations and Matrices
Michael L O'Leary · 2021
Transformations are good to use when the elements of the space need to be moved, but if the function also needs to take into consideration any algebra of the space, not just any transformation will do. The subject linear algebra can be considered the study of linear transformations. The purpose of matrices and matrix multiplication is to represent linear transformations. The initial definition of matrix multiplication was motivated by an attempt to describe a linear transformation. This suggests that if an entire algebraic system is to be defined on matrices like that defined on the real numbers, it should flow from the study of linear transformations. A reflection through a line containing the zero vector is a linear operator. Gauss-Jordan elimination will lead to the reduced row echelon form of the coefficient matrix.