The Algebra of Matrices
This chapter defines the vector operations on matrices, namely, the addition of two matrices, and the multiplication of a matrix by a scalar. There are many ways of mapping a plane into itself linearly; that is, so that any linear combination of vectors is carried into the same linear combination of transformed vectors. The algebra of linear transformations (matrices) involves three operations: addition of two linear transformations (or matrices), multiplication of a linear transformation by a scalar, and multiplication of two linear transformations (matrices). The most important combination of two linear transformations T and U is their product TU. The chapter considers only the product of two linear transformations T, U of a vector space V into itself. Any group of linear transformations may be represented by a corresponding group of matrices. Linear transformations of a finite-dimensional vector space are of two kinds: either bijective (both one-one and onto) or neither injective nor surjective (neither one-one nor onto).