Liouville, Joseph 317

p

–

q

Any linear transformation Tfrom a vector

space Vto a vector space Wis given by multi-

plication with a matrix Awhose jth column is

the effect of Ton the jth basis vector of V.

For example, consider a rotation Rin the plane

about the origin through an angle θ. Such a rotation

takes a unit vector in the direction of the x-axis,

, to the vector , and the unit vector in

the direction of the y-axis, , to the vector

. Thus the matrix representing a rotation

through an angle θis given by:

In the same way, a reflection about the x-axis, for

instance, is given by the matrix:

and a dilation by a factor kas:

Of course, if one were to work with a different set of

basis vectors, the matrix representing the linear trans-

formation would be different.

One can show that if matrix Arepresents a linear

transformation T:V →Wand matrix Brepresents a

linear transformation S:W →R, then the matrix prod-

uct BA represents the composite linear transformation:

SoT:V →R. If the matrix Ais invertible, then the

INVERSE MATRIX

A–1 represents the inverse linear trans-

formation T–1:W→V. (This inverse map exists if Ais

indeed invertible.)

See also

AFFINE TRANSFORMATION

;

MATRIX OPER

-

ATIONS

.

Liouville, Joseph (1809–1882) French Number the-

ory, Analysis Born on March 24, 1809, in Saint-

Omer, France, scholar Joseph Liouville is best

remembered for his 1844 proof of the existence of

TRANSCENDENTAL NUMBER

s. Liouville also managed to

provide, for the first time, specific examples of numbers

that cannot be algebraic. (These numbers are today

called Liouville numbers.) He is also noted for his con-

tributions to

DIFFERENTIAL EQUATION

s, differential

geometry (the study of

CALCULUS

on three-dimensional

shapes and surfaces), complex analysis (calculus

applied to complex numbers), and

NUMBER THEORY

. In

1864 he also edited and published manuscripts left by

É

VARISTE

G

ALOIS

(1811–32) on

POLYNOMIAL

equa-

tions. Liouville wrote over 400 mathematical papers

during his career, around 200 of which were on the

topic of number theory.

Liouville graduated from the École Polytechnique

in 1827 with a basic degree in mathematics and

mechanics. After taking on a number of different teach-

ing positions, Liouville was eventually appointed pro-

fessor of analysis and mechanics at that same

institution in 1838. Meanwhile, Liouville had already

garnered an international reputation for his work on

electrodynamics, partial differential equations, and the

study of heat, as well as for his establishment of a new

mathematics journal, Journal de Mathématiques Pures

et Appliqués (Journal of pure and applied mathemat-

ics), today commonly referred to as Liouville’s Journal.

Correspondence with mathematicians C

HRISTIAN

G

OLDBACH

and Daniel Bernoulli of the B

ERNOULLI

FAMILY

sparked Liouville’s interest in transcendental

numbers. He attempted to prove that the number ewas

transcendental, but did not succeed. (This feat was

later accomplished by French mathematician Charles

Hermite in 1873.) However, using the theory of contin-

ued fractions, Liouville managed to construct a class of

real numbers xwith the property that for each natural

number nthere is a fraction satisfying the inequality:

This, Liouville showed, was enough to establish that x

is transcendental. In particular, Liouville showed that

the specific number (sometimes now called Liouville’s

number):