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 .

Then

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

or 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:

Mon Aug 24 16:30:59 EDT 1998