It is useful to formalize the operations on vectors and matrices that form the basis of linear algebra. For our purpose, the most useful definitions are the following.
A matrix is a rectangular array of numbers written in the form
The matrix A has dimensions if it has m rows and n columns. When m=1, the matrix is called a row vector; when n=1, the matrix is called a column vector. A vector can be represented either by a row vector or a column vector.
Equality of two matrices of same dimensions:
Let and .
Then A=B means that for all i,j.
Multiplication of a matrix A by a scalar k:
Addition of two matrices of same dimensions:
Let and .
Note that A+B is not defined when A and B have different dimensions.
Multiplication of a matrix of dimensions by a matrix of dimensions :
Let and .
Then AB is a matrix of dimensions computed as follows.
As an example, let us multiply the matrices
The result is
Note that AB is defined only when the number of columns of A equals the number of rows of B. An important remark: even when both AB and BA are defined, the results are usually different. A property of matrix multiplication is the following:
That is, if you have three matrices A, B, C to multiply and the product is legal (the number of columns of A equals the number of rows of B and the number of columns of B equals the number of rows of C), then you have two possibilities: you can first compute AB and multiply the result by C, or you can first compute BC and multiply A by the result.
Remark: A system of linear equations can be written conveniently using matrix notation. Namely,
can be written as
So a matrix equation Ax = b where A is a given matrix, b is a given m-column vector and x is an unknown n-column vector, is a linear system of m equations in n variables. Similarly, a vector equation where are given m-column vectors and are n unknown real numbers, is also a system of m equations in n variables.
The following standard definitions will be useful: