Talk:Dual space (linear algebra)

An interesting one-dimensional example, showing very clearly the difference between the space and its dual, is: the dual to the time axis is the frequency axis. --Boris Tsirelson 11:51, 22 January 2011 (EST)

Don't you need a Fourier transform to see it?--Paul Wormer 12:14, 22 January 2011 (EST)

Probably I saw it in a book about Fourier transform and uncertainty principle. (It was even written somewhere that a musical note is roughly a pair (time,frequency)...) But still, I believe that this can be explained to any musician. Consider something that oscillate in time, with a frequency. The number of oscillations (already made) is a function of time. Counting also parts of oscillations we get a linear function of time, just (frequency)×(time). In this sense, the frequency parametrizes linear functionals on the time axis. Sure, a man acquainted with Fourier transform will recognize it. But a man unacquainted with it still can understand the idea. You see, the dimension of frequency is, (time)^-1. --Boris Tsirelson 14:04, 22 January 2011 (EST)

This seems to me more of "reciprocal space" defined by the aid of an inner product on the original space. But of course, the dual (by linear functionals) and the reciprocal (by inner products) space can be identified, and their distinction is moot. In any case, the argument of an exponential is dimensionless, thus νt, k⋅r, etc. are all dual/reciprocal pairs.--Paul Wormer 04:02, 23 January 2011 (EST)

Inner product, why? I do not need any inner product on the time axis (nor the frequency axis); only the bilinear pairing between the two. My description intentionally does not use any units. The number of oscillations is dimensionless, and clearly visible irrespective of any measurement units. --Boris Tsirelson 07:38, 23 January 2011 (EST)

Let x be the time from my birth till now (not a number, just a vector on the time axis). Let y be the frequency of Earth rotation around its axis (not a number, just a vector on the frequency axis). Then xy is a number, about 20000. --Boris Tsirelson 08:04, 23 January 2011 (EST)

[Edit conflict, I wrote the following before your second comment]

Well, ν and t being real, you could formally call the product νt an inner product on R, just like k⋅r is on R³. I also don't talk about units, but about dimensions, as you do when you write: " You see, the dimension of frequency is, (time)^-1." And you're quite right: the "number of oscillations" is meaningless unless you add: "per time interval". --Paul Wormer 08:20, 23 January 2011 (EST)

To your second comment: how can you say that your age is a vector and not a number? To me that is incomprehensible. --Paul Wormer 08:20, 23 January 2011 (EST)

Surely not a number. Let a be one second, then x=2000000000a (approximately); 2000000000 is a number, but x is not. By the way: today's "Euclidean geometry" is not really Euclidean. Euclid never told that a number corresponds to every line segment. He told that a number can express the relation between two segments.

Every number is either less than 7, or equal to 7, or more than 7. But my age could not be less (nor more) than 7. It is now less than 7 ages of the Earth, and greater than 7 days.

Continuing the example: in contrast, xx is not a number; no inner product given on the time axis (unless/until you choose a unit). --Boris Tsirelson 09:21, 23 January 2011 (EST)

[unindent] It is probably better to stop this discussion. It would be much more amusing with a drink in a bar than on the internet. (Although I insist that νt, being dimensionless, is a number, invariant under choice of unit. You count the number of oscillations in time t and this number does not depend on how you express t, in picoseconds, years, or lifetimes of the universe. If νt were not dimensionless, you could not compute exp[iνt].) --Paul Wormer 10:52, 23 January 2011 (EST)

But then, where is the discussion? It seems, we both agree that dimensionless thing (and only such) is a number! Your νt is mine xy. But confusingly you wrote that my age is a number, while it is of dimension (time). --Boris Tsirelson 11:11, 23 January 2011 (EST)

It was not for the discussion, but for possible profit to the article. --Boris Tsirelson 11:13, 23 January 2011 (EST)

I've always thought of age as a scalar, because it is the number of days, years, or time-units something has been in existence, and (for nearly all purposes, until time-travel is invented) ;-) time only moves "forward"—so direction is irrelevant. At least, I've never seen time used as a vector.—Tom Larsen (talk) 23:58, 23 January 2011 (EST)

OK, this is also true. I call it vector here only for treating the one-dim case as a special case of the n-dim case, which is completely legitimate (at least, mathematically). I emphasize here not the distinction between dim 1 and higher dim, but the distinction between Euclidean space (with a given metric) and linear space (no metric given), which in the one-dim case becomes almost the same as the distinction between dimensionless (in the physical space) and having nontrivial physical dimension. --Boris Tsirelson 00:53, 24 January 2011 (EST)

About time travel. Translated into math language it means that this one-dim space is oriented. Orientation applies also to n-dim and is more complicated notion, but in 1-dim it just chooses one of the two directions.

The positive side of our mutual non-understanding is the idea of writing an article that explains just these, rather basis, conceptions, misconceptions and language/cultural distinction between math, physics etc. Mathematicians know that, paradoxically, simpler (say, 1-dim) cases are often harder for non-mathematicians. I recall entry exam in a univ in Russia, where the question "multiply 5 by 0" was crashing. --Boris Tsirelson 01:35, 24 January 2011 (EST)

I try to volunteer something in this direction; please look: User:Boris Tsirelson/Sandbox1. I hesitate to do it in the mainspace, probably because I still treat it as a city rather than building ground. However, I do seek massive collaboration. If you find it a good idea, create something like this in the mainspace. --Boris Tsirelson 02:08, 24 January 2011 (EST)