Re: Isomorphism of vector spaces (preserves dimension) -- a proof ?
MeAmI.org
qsymmetry wrote:
> > If f : V -> W is an isomorphism of vector spaces, why
> > does it follow that dim V = dim W?
> >
> > Here, we do not assume that V, W are finite
> > dimensional.
>
> I argued as follows:
>
>
>
> Suppose B is a basis for V; then f(B) is a linearly independent subset which spans W,
> since f is surjective; hence f(B) is a basis for W.
>
> Since f is injective, we can identify B with its image f(B), i.e. |B|= |f(B)|, in which case
>
>
> dim V = |B| = |f(B)| = dim W.
>
>
> Does this work?
Yes and it is genius. I compliment you on the flexible identity you
chose. I believe this is a smart thing to do since it allows more
options without imposing any restrictions when none are necessary to
prove the bijection.
Again, great work. Thank you for this contribution.
Sincerely,
Martin Musatov
http://MeAmI.org
"It's better than Google Search and I'm just saying it because it's
true."
--
[ ]
P.S. It is funny now "Dim" is a variable like "x" or "y" in Algebra.
In all seriousness though this is such an interesting trait I must
express it is the coming of a new mathematics. One in which sensible
meanings can be advancingly refined to computable variables of single
letters than imagine the potential of a machine to juggle multi-level
complex compound variables. What I am referring to is more than loose
grammars here. A machine is not intelligent. It is automated. Combine
this with Edison's classical observation that the only capital a human
being really has is time. It's the one thing the one thing we never
have enough of and when it is gone we can never get it back. Combime
this further the opinion the only thing any machine has on a person is
time and strength. A computer can conisder many combinations much more
quickly than a person. As a person can produce energy required to walk
so can a gallon of petroleum push a metal wagon down a street. The
petroleum provides the energy for the wagon to move but it still
requires steering. I maintain such is the case with computation. The
electricity through the silicon provides the power to flag a million
combinations of symbolic identities but the symbolic entities in and
of themselves do not think. As a machine is powered by petroleum a
computer is powered by electricity. The product of the result is
dependent on intervention only possible sourced from thought of a
human being. This is true so much as to our benefits and perception of
these physical consequence. As to the old question if a tree falls in
the forest and no one is around to hear it does it make a sound? If a
machine chooses the right answer and there is no one there to confirm
does it think?
The answer is no. The solution is proof P==NP.
Martin Musatov
P.S.^2 I apologize to the author for "highjacking" this thread. As
quizically as inspiration approaches a writer even a human has not
refined. Perhaps this is a problem for machines. What do you think?