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Topic: Matrices: If AB = I, then BA = I.
Replies: 52   Last Post: Apr 24, 1999 10:56 PM

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Bill Dubuque

Posts: 156
Registered: 12/6/04
Re: The infinite hotel, shifts, 1-1 iff onto, pigeonhole principle
Posted: Apr 24, 1999 6:30 PM
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The Hilbert Hotel is a classical popular realization of the first
infinite ordinal w. It's a nice example of the dichotomy between
the finite and the infinite: the right-shift operator n -> n+1
is 1-1 but not onto the infinite set w; contrast this with a
finite set, where a function is 1-1 if and only if it is onto
(the pigeonhole principle). The exact same contrast occurs between
finite and infinite dimensional vectors spaces and can again be
illustrated via shift operators, e.g. if V has infinite dimension

let R = (Right) shift operator: R(a b c d ...) = (0 a b c ...)
and L = (Left) shift operator: L(a b c d ...) = (b c d e ...)
so LR = I but RL != I since: RL(a b c d ...) = (0 b c d ...)

This has a simple model in a vector space V of polynomials. Let V
have basis {1, x, x^2 ...}, so (a b c ...) = a + b x + c x^2 + ...
with shift operators R p(x) = x p(x), L p(x) = 1/x (p(x)-p(0))
so that LR = I but RL != I since RL p(x) = p(x)-p(0). Note

L is onto but not 1-1, since L(Rp)=p but L(1)=0.

R is 1-1 but not onto; its image RV = xV is a proper

subspace isomorphic to V (since Im R ~= V/(Ker R) = V/0 = V);
note: dim V is infinite iff V is isomorphic to a proper subspace,
just as a set is infinite iff it's isomorphic to a proper subset.

In fact this equivalence between 1-1 and onto maps comes from
a general pigeonhole principle in lattices - in particular it
holds for any algebraic structure of finite height (in its
lattice of subalgebras) -- see my soon to appear post [1].

The shift operator is of fundamental importance in linear algebra,
e.g. see the review of Fuhrmann's book in my prior post [2].

The hotel is usually called "Hilbert's Hotel" - named after the great
mathematician David Hilbert, who often mentioned it in his popular
lectures; cf. Rucker: Infinity and the Mind p. 73, where it is also
mentioned that the Polish science fiction writer Stanislew Lem once
wrote a short story about Hilbert's Hotel, which appears in Vilenkin's
book Stories About Sets. Rudy Rucker's book is one of the best popular
expositions of most all aspects of infinity - highly recommended [3] [4].

-Bill Dubuque

[1] http://www.dejanews.com/dnquery.xp?QRY=dubuque%20fuhrmann%20holder&groups=sci.math&ST=PS
[2] http://www.dejanews.com/dnquery.xp?QRY=dubuque%20fuhrmann&groups=sci.math&ST=PS
[3] http://www.dejanews.com/getdoc.xp?AN=437551805
[4] http://www.dejanews.com/dnquery.xp?QRY=dubuque%20rucker&groups=sci.math%2A%20sci.logic&ST=PS






Date Subject Author
3/13/99
Read Matrices: If AB = I, then BA = I.
JRic395488
3/14/99
Read flawed 'proof'
Chan-Ho
3/14/99
Read Re: flawed 'proof'
John Creighton
3/14/99
Read Re: flawed 'proof'
David C. Ullrich
3/14/99
Read Re: flawed 'proof'
David C. Ullrich
3/15/99
Read Re: flawed 'proof'
Horst Kraemer
3/14/99
Read Re: Matrices: If AB = I, then BA = I.
Hope Hubris
3/14/99
Read Re: Matrices: If AB = I, then BA = I.
JRic395488
3/17/99
Read Re: Matrices: If AB = I, then BA = I.
David Kastrup
3/19/99
Read Re: Matrices: If AB = I, then BA = I.
Erland Gadde
3/15/99
Read Re: Matrices: If AB = I, then BA = I.
bobs@rsa.com
3/15/99
Read Re: Matrices: If AB = I, then BA = I.
Ake Brannstrom
3/16/99
Read Re: Matrices: If AB = I, then BA = I.
Paul Hammond
3/16/99
Read Re: Matrices: If AB = I, then BA = I.
Mike ROBSON
3/15/99
Read Re: Matrices: If AB = I, then BA = I.
Christian Ohn
3/15/99
Read Re: Matrices: If AB = I, then BA = I.
David C. Ullrich
3/16/99
Read Re: Matrices: If AB = I, then BA = I.
Christian Ohn
3/18/99
Read Re: Matrices: If AB = I, then BA = I.
David C. Ullrich
3/18/99
Read Re: Matrices: If AB = I, then BA = I.
Christian Ohn
3/19/99
Read Re: Matrices: If AB = I, then BA = I.
David C. Ullrich
3/18/99
Read Re: Matrices: If AB = I, then BA = I.
Zdislav V. Kovarik
3/19/99
Read Re: Matrices: If AB = I, then BA = I.
David C. Ullrich
3/20/99
Read Re: Matrices: If AB = I, then BA = I.
David C. Ullrich
3/17/99
Read Re: Matrices: If AB = I, then BA = I.
D. J. Bernstein
3/17/99
Read Re: Matrices: If AB = I, then BA = I.
JRic395488
3/19/99
Read Re: Matrices: If AB = I, then BA = I.
David C. Ullrich
3/18/99
Read Re: Matrices: If AB = I, then BA = I.
David C. Ullrich
3/19/99
Read Re: Matrices: If AB = I, then BA = I.
David C. Ullrich
3/15/99
Read Re: Matrices: If AB = I, then BA = I.
bobs@rsa.com
3/15/99
Read Re: Matrices: If AB = I, then BA = I.
David Kastrup
3/15/99
Read Re: Matrices: If AB = I, then BA = I.
David C. Ullrich
3/16/99
Read Re: Matrices: If AB = I, then BA = I.
bobs@rsa.com
3/15/99
Read Re: Matrices: If AB = I, then BA = I.
Richard Carr
3/15/99
Read Re: Matrices: If AB = I, then BA = I.
David C. Ullrich
3/15/99
Read Re: Matrices: If AB = I, then BA = I.
Lee Rudolph
3/16/99
Read Re: Matrices: If AB = I, then BA = I.
David C. Ullrich
3/16/99
Read Re: Matrices: If AB = I, then BA = I.
Timothy Murphy
3/17/99
Read Re: Matrices: If AB = I, then BA = I.
John Harper
3/16/99
Read Re: Matrices: If AB = I, then BA = I.
Keith Ramsay
3/18/99
Read Re: Matrices: If AB = I, then BA = I.
D. J. Bernstein
3/19/99
Read Re: Matrices: If AB = I, then BA = I.
Erland Gadde
3/19/99
Read Re: Matrices: If AB = I, then BA = I.
Charles H. Giffen
3/15/99
Read Re: Matrices: If AB = I, then BA = I.
JRic395488
3/15/99
Read Re: Matrices: If AB = I, then BA = I.
Doug Norris
3/19/99
Read Re: Matrices: If AB = I, then BA = I.
Erland Gadde
3/15/99
Read Re: Matrices: If AB = I, then BA = I.
Jim Ferry
3/15/99
Read Re: Matrices: If AB = I, then BA = I.
Robert J. Kolker
3/17/99
Read AB = I implies BA = I
Timothy P. Keller
3/19/99
Read Re: Matrices: If AB = I, then BA = I.
Erland Gadde
3/21/99
Read Re: Matrices: If AB = I, then BA = I.
Robert J. Kolker
3/21/99
Read Re: Matrices: If AB = I, then BA = I.
Alexander Abian
4/24/99
Read Re: The infinite hotel, shifts, 1-1 iff onto, pigeonhole principle
Bill Dubuque
4/24/99
Read Re: The infinite hotel, shifts, 1-1 iff onto, pigeonhole principle
Andrew Martin

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