Talk:Vector space: Difference between revisions
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I agree, and have attempted to add some introductory paragraphs, although I am finding it tricky to be precise but non-technical. | I agree, and have attempted to add some introductory paragraphs, although I am finding it tricky to be precise but non-technical. | ||
If you have any ideas or suggestions, please add them! [[User:Michael Underwood|Michael Underwood]] 13:01, 25 July 2007 (CDT) | If you have any ideas or suggestions, please add them! [[User:Michael Underwood|Michael Underwood]] 13:01, 25 July 2007 (CDT) | ||
== Scalars == | |||
I don't think the word "field" should appear on the main vector space page, but rather on an advanced version of the page. If it must be included, why not just go all the way to "division ring"? Many people need or want to know about vector spaces without ever needing to know about abstract fields. Questions: | |||
* Is there general agreement on this? | |||
* Which scalars should be discussed? Reals only, reals and complexes, or even reals, complexes and GF(2)? | |||
I personally like defining things in terms of real scalars initially and through the bulk of the discussion, then adding a section about vector spaces over the complex numbers or over GF(2).[[User:Barry R. Smith|Barry R. Smith]] 22:29, 25 November 2008 (UTC) | |||
== V prime: do you see it? == | |||
On my browser "V prime" looks practically the same as "V". Is it the case on other browsers? (Look at Section 3.2 "dimension".) [[User:Boris Tsirelson|Boris Tsirelson]] 09:10, 24 July 2009 (UTC) | |||
:I use FireFox and see it perfectly (it is even possible that I'm the one who introduced ''V′'' ). If you wish you can change it to, say, ''W'', or ''U''.--[[User:Paul Wormer|Paul Wormer]] 10:52, 24 July 2009 (UTC) | |||
::OK, it is my problem (of my new browser, Iceweasel on Debian Linux); and indeed, Safari on MacOS works fine. [[User:Boris Tsirelson|Boris Tsirelson]] 11:19, 24 July 2009 (UTC) | |||
== Remarks == | |||
"A vector space V over a field F is a set that satisfies certain axioms (see below) and which is equipped with two operations" — operations satisfy axioms, not the set. Or at least, all together (set and operations) satisfies axioms. | |||
"The vector \vec{u}+\vec{v} is also an element of V. This is automatically satisfied when the addition operation is defined as being injective as it was above" — as far as I know, "injective" means something different from what is meant here. | |||
"7. Scalar multiplication is distributive over addition in F" — strangely, a link to "distributive" appears only later, in (8). | |||
"In general there are infinitely many linearly independent vectors in a vector space" — "in general" is not apt, I'm afraid. | |||
"If V′ is a linear subspace of the n-dimensional space V (all elements of V′ belong simultaneously to V ), and V′ contains a set B of m linearly independent vectors then m < n" — no, it may be m=n. | |||
"Hence it is not possible to find a fifth vector linearly independent of the vectors (1): any five vectors form a linearly dependent set" — So simple? The second phrase is not an immediate consequence of the former, if I am not mistaken. A finer argument is needed. | |||
"Applications of vector spaces" — for now it is better to remove this empty section. | |||
[[User:Boris Tsirelson|Boris Tsirelson]] 15:40, 25 May 2010 (UTC) | |||
Latest revision as of 09:40, 25 May 2010
Perhaps there needs to be a bit more introduction? Natalie Watson 14:58, 13 July 2007 (CDT)
I agree, and have attempted to add some introductory paragraphs, although I am finding it tricky to be precise but non-technical. If you have any ideas or suggestions, please add them! Michael Underwood 13:01, 25 July 2007 (CDT)
Scalars
I don't think the word "field" should appear on the main vector space page, but rather on an advanced version of the page. If it must be included, why not just go all the way to "division ring"? Many people need or want to know about vector spaces without ever needing to know about abstract fields. Questions:
- Is there general agreement on this?
- Which scalars should be discussed? Reals only, reals and complexes, or even reals, complexes and GF(2)?
I personally like defining things in terms of real scalars initially and through the bulk of the discussion, then adding a section about vector spaces over the complex numbers or over GF(2).Barry R. Smith 22:29, 25 November 2008 (UTC)
V prime: do you see it?
On my browser "V prime" looks practically the same as "V". Is it the case on other browsers? (Look at Section 3.2 "dimension".) Boris Tsirelson 09:10, 24 July 2009 (UTC)
- I use FireFox and see it perfectly (it is even possible that I'm the one who introduced V′ ). If you wish you can change it to, say, W, or U.--Paul Wormer 10:52, 24 July 2009 (UTC)
- OK, it is my problem (of my new browser, Iceweasel on Debian Linux); and indeed, Safari on MacOS works fine. Boris Tsirelson 11:19, 24 July 2009 (UTC)
Remarks
"A vector space V over a field F is a set that satisfies certain axioms (see below) and which is equipped with two operations" — operations satisfy axioms, not the set. Or at least, all together (set and operations) satisfies axioms.
"The vector \vec{u}+\vec{v} is also an element of V. This is automatically satisfied when the addition operation is defined as being injective as it was above" — as far as I know, "injective" means something different from what is meant here.
"7. Scalar multiplication is distributive over addition in F" — strangely, a link to "distributive" appears only later, in (8).
"In general there are infinitely many linearly independent vectors in a vector space" — "in general" is not apt, I'm afraid.
"If V′ is a linear subspace of the n-dimensional space V (all elements of V′ belong simultaneously to V ), and V′ contains a set B of m linearly independent vectors then m < n" — no, it may be m=n.
"Hence it is not possible to find a fifth vector linearly independent of the vectors (1): any five vectors form a linearly dependent set" — So simple? The second phrase is not an immediate consequence of the former, if I am not mistaken. A finer argument is needed.
"Applications of vector spaces" — for now it is better to remove this empty section.
Boris Tsirelson 15:40, 25 May 2010 (UTC)