Re: combining vector spaces

In article <1125804408.230235.229010@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
Edward Green <spamspamspam3@xxxxxxxxxxx> wrote:
>First, I apologize for asking an actual basic question about physics.
>I realize this is poor netiquette, but I'm a newbie and don't know any
>better!

Hello, Edward, you newbie. You'll learn in time that you're not supposed
to ask real questions here.

>What I've been trying to understand is the various ways vector
>spaces can be combined to make new vector spaces. I suspect the
>phrases "direct product" and "direct sum", and possibly "outer product"
>are going to come up.
>
>Say we have a vector space and a handy set of n basis vectors, and
>somebody comes along and says "Hi. I've got these two extra basis
>vectors which nobody knew about before, but should really have been
>included in your kit. I found them on the cutting room floor, and I
>thought you'd like to have them".
>
>Ok. So we glom these extra vectors onto our space, so that now all our
>n-tuples become n+2-tuples. Now, we notice that these two extra basis
>vectors, which are orthogonal to all the ones we are currently using,
>span a perfectly good space all on their own; so we are combining
>vector spaces.
>
>These vector spaces are blood-type compatible, so they combine pretty
>smoothly; we find we can rotate vectors out of one space into the other
>continuously -- and stop at all projections in between, with shadows in
>both sub-spaces. And we note it is neither necessary nor sufficient
>that one of the component inner products is zero for the total inner
>product of vectors living in the larger space to be zero. Like adding
>memory to a Mac, it's seemless, and the machine knows how to use it.
>
>But now, some other troublemaker comes along and hands us two more
>basis vectors he wants us to include in our space; but these ones are
>different from the ones we have! The're like oil and water, they live
>on another planet, speak a different language, and have nothing
>obviously to do with the vectors we already have. But this dictator
>tells us that the comrades are here to help us, and that the new rule
>of inner products is not to add the component inner products, but to
>_multiply_ them. So now it _is_ both necessary and sufficient that at
>least one of the sub-inner products be zero in order for the combined
>inner product to be zero.
>
>We scratch our heads, but are forced to accept this, even though these
>new vectors have no apperent relation to those we already have: the're
>like shadow vectors in another world.
>
>Well, I think I've described the possibilities fairly, and even could
>give some examples, but I don't really grok the meaning of the last
>possibility, or why we want/are forced to use it. Please help the
>understanding homeless. Give generously.
>

All I know from quantum mechanics is that if we have a Hilbert space U
with the basis |u> and another one V with the basis |v>, and we want a
wavefunction with both of them, we say that we take the Cartesian product
U x V with the basis |u> (x) |v> = |u,v>, and stop asking silly questions.
Actually, I think most physicists stop asking silly questions before they
get to the Cartesian product part, and say that if there are two variables
they'll just call the vector |u,v>, because that has two indices. I
suppose there must be some condition on that, like |u> and |v> not
mixing.

Have you found a way in which that physicist-level procedure is inadequate
and mistaken that for a stupid question?


--
"We've all heard that a million monkeys banging on a million typewriters
will eventually reproduce the entire works of Shakespeare. Now, thanks to
the Internet, we know this is not true." -- Robert Wilensky
.

Relevant Pages

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    (sci.physics)
  • Re: Test SPF3
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  • Re: combining vector spaces
    ... I apologize for asking an actual basic question about physics. ... > that one of the component inner products is zero for the total inner ... > product of vectors living in the larger space to be zero. ... > basis vectors he wants us to include in our space; ...
    (sci.physics)
  • Re: Inner product spaces
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  • Re: inner products, vector norms and matrix norms
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