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# Operations on Vectors and Matrices

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:

Next: Linear Combinations Up: Basic Linear Algebra Previous: Linear Equations

Michael A. Trick
Mon Aug 24 16:30:59 EDT 1998