500 Chapter 18Matrices and linear transformations
In this notation, the elementa
ijlies in the ith row and jth column. Although curved
brackets are used in this text, matrices are often written with square brackets, and are
also denoted by symbols such as(a
ij)and[a
ij]. To qualify as matrices, arrays must
obey the laws of matrix algebra (Section 18.3).
Matrix algebra has its origins in the description of coordinate transformations.
2Let
the pair of linear equations
x′ 1 = 1 a
1x 1 + 1 b
1y
(18.3)
y′ 1 = 1 a
2x 1 + 1 b
2y
describe a change of coordinates in the xy-plane from(x, y)to(x′, y′). This coordinate
transformationis completely characterized by the four coefficientsa
1, b
1, a
2,b
2; that
is, by the array
(18.4)
In matrix algebra, the equations (18.3) are written as the single matrix equation
(18.5)
or, assigning symbols to the arrays,
r′ 1 = 1 Ar (18.6)
where
(18.7)
The quantities rand r′are column matrices(or column vectors) whose elements
are the coordinates before and after the coordinate transformation, and Ais the
square matrixof the coefficients that represents and determines the transformation.
Coordinate transformations are examples of the linear transformations discussed in
Section 18.5.
rr A=
,′=
′
′
,=
x
y
x
y
ab
ab
1
122x
y
ab
ab
x
y
′
′
=
1
122ab
ab
1
1222The algebra of coordinate transformations was discussed by Gauss in 1801, and developed into an algebra by
Cayley in his Memoir on the theory of matricesof 1858, in which he introduced the single-letter notation for a
matrix and derived the rules of addition and multiplication.
Arthur Cayley (1821–1895), graduated from Trinity College, Cambridge, in 1842, was Fellow for seven years,
then practiced law for 14 years during which time he published several hundreds of papers. His output is rivalled
only by Euler and Cauchy, with 967 published papers in his Collected Works. Best known for his work on matrices
and forms, he also developed the analytical geometry of n-dimensional space in 1843, gave the first definition of
an abstract group in 1854, and made contributions to graph theory, with an application to the study of chemical
isomers in 1874. He was in great demand as a referee because of his encyclopaedic knowledge of mathematics.