Page 17 - A Course in Linear Algebra with Applications

Chapter One

MATRIX ALGEBRA

In this first chapter we shall introduce one of the prin-
cipal objects of study in linear algebra, a matrix or rectan-
gular array of numbers, together with the standard matrix
operations. Matrices are encountered frequently in many ar-
eas of mathematics, engineering, and the physical and social
sciences, typically when data is given in tabular form. But
perhaps the most familiar situation in which matrices arise is
in the solution of systems of linear equations.

1.1 Matrices

An m x n matrix A is a rectangular array of numbers,
real or complex, with m rows and n columns. We shall write
dij for the number that appears in the ith row and the jth
column of A; this is called the (i,j) entry of A. We can either
write A in the extended form

/ a n &12 • • • CL\ n \
«22 - - ' &2n
«21
Q"m2 ' ' ' Q"mn '
V&rol
or in the more compact form

Thus in the compact form a formula for the (i,j) entry of A
is given inside the round brackets, while the subscripts m and
n tell us the respective numbers of rows and columns of A.

1