Any linear transformation T from a vector
space V to a vector space W is given by multiplication with a matrix A whose jth column is
the effect of T on the jth basis vector of V.
For example, consider a rotation R in 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 θ is 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 k as:
Of course, if one were to work with a different set of
basis vectors, the matrix representing the linear transformation would be different.
One can show that if matrix A represents a linear
transformation T:V → W and matrix B represents a
linear transformation S:W → R, then the matrix product BA represents the composite linear transformation:
S o T:V → R. If the matrix A is invertible, then the
INVERSE MATRIX A
–1 represents the inverse linear transformation T
–1 :W → V. (This inverse map exists if A is
indeed invertible.)
See also AFFINE TRANSFORMATION; MATRIX OPERATIONS.
Liouville, Joseph (1809–1882) French Number theory, Analysis Born on March 24, 1809, in SaintOmer, France, scholar Joseph Liouville is best
remembered for his 1844 proof of the existence of
TRANSCENDENTAL NUMBERs. 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 contributions to DIFFERENTIAL EQUATIONs, 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
ÉVARISTE GALOIS (1811–32) on POLYNOMIAL equations. 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 teaching positions, Liouville was eventually appointed professor 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 mathematics), today commonly referred to as Liouville’s Journal.
Correspondence with mathematicians CHRISTIAN
GOLDBACH and Daniel Bernoulli of the BERNOULLI
FAMILY sparked Liouville’s interest in transcendental
numbers. He attempted to prove that the number e was
transcendental, but did not succeed. (This feat was
later accomplished by French mathematician Charles
Hermite in 1873.) However, using the theory of continued fractions, Liouville managed to construct a class of
real numbers x with the property that for each natural
number n there 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):
|
|
x
p
q
q
n
−
<
1
p
– q
k
k
0
0
1 0
0 1
−
R =
−
cos
sin
sin
cos
θ
θ
θ
θ
– sin
cos
θ
θ
e 2
0
1
=
cos
sin
θ
θ
e 1
1
0
=
Liouville, Joseph 317
space V to a vector space W is given by multiplication with a matrix A whose jth column is
the effect of T on the jth basis vector of V.
For example, consider a rotation R in 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 θ is 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 k as:
Of course, if one were to work with a different set of
basis vectors, the matrix representing the linear transformation would be different.
One can show that if matrix A represents a linear
transformation T:V → W and matrix B represents a
linear transformation S:W → R, then the matrix product BA represents the composite linear transformation:
S o T:V → R. If the matrix A is invertible, then the
INVERSE MATRIX A
–1 represents the inverse linear transformation T
–1 :W → V. (This inverse map exists if A is
indeed invertible.)
See also AFFINE TRANSFORMATION; MATRIX OPERATIONS.
Liouville, Joseph (1809–1882) French Number theory, Analysis Born on March 24, 1809, in SaintOmer, France, scholar Joseph Liouville is best
remembered for his 1844 proof of the existence of
TRANSCENDENTAL NUMBERs. 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 contributions to DIFFERENTIAL EQUATIONs, 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
ÉVARISTE GALOIS (1811–32) on POLYNOMIAL equations. 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 teaching positions, Liouville was eventually appointed professor 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 mathematics), today commonly referred to as Liouville’s Journal.
Correspondence with mathematicians CHRISTIAN
GOLDBACH and Daniel Bernoulli of the BERNOULLI
FAMILY sparked Liouville’s interest in transcendental
numbers. He attempted to prove that the number e was
transcendental, but did not succeed. (This feat was
later accomplished by French mathematician Charles
Hermite in 1873.) However, using the theory of continued fractions, Liouville managed to construct a class of
real numbers x with the property that for each natural
number n there 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):
|
|
x
p
q
q
n
−
<
1
p
– q
k
k
0
0
1 0
0 1
−
R =
−
cos
sin
sin
cos
θ
θ
θ
θ
– sin
cos
θ
θ
e 2
0
1
=
cos
sin
θ
θ
e 1
1
0
=
Liouville, Joseph 317