Talk:Multivector

message: I've just changed "A sum of only n-grade components is called an n-vector, or a homogeneous k-blade." to "A sum of only n-grade components is called an n-vector, or a homogeneous multivector." All other sources I've seen, incliding the reference cited here, describe blades are *simple* multivectors, i.e. a scalar or a wedge product of some number of the generating vectors of the algebra, rather than a sum of wedge products. Dependent Variable (talk) 11:43, 14 September 2010 (UTC)
thread_title: Blade

message: The lede has become confusing. The statement "A multivector that is a linear combination of basis multivectors constructed from p basis elements is called a rank p multivector, or a p-vector" is difficult to interpret in the way it is intended (what is "constructed from"?).  "A vector space with a vector multiplication is called an algebra" is unnecessarily imprecise. The term "vector space" is used in the lede to describe different subspaces of the algebra without highlighting the distinction.  Use of the term "rank" is inconsistent with elsewhere in WP, most relevantly with Wedge product#Rank of a k-vector); we use the term "grade" in this context.  Including illustrations of the wedge product in the body is unnecessary duplication; I would not like to see WP become an endless series of repetitive tutorials: Wikipedia is not a textbook. Although the article was and still is sorely incomplete, the recent changes have not, IMO, been an improvement. —Quondum 17:24, 26 March 2014 (UTC)
replies: text: For what it's worth here is the comparison: . M∧Ŝc2ħεИτlk 17:33, 26 March 2014 (UTC)
replies: text: Perhaps part of the problem is that the term is very context-specific – i.e., it is not a standalone concept.  The lead should first establish the context, which is exterior algebra or geometric algebra.  The basic definition is as a general element of the algebra (i.e. not as built up in any independent way).  The article should not try to define these algebras, but it should highlight that multivectors are linear combination of k-vectors of different k, which in turn are linear sums of k-blades (i.e. k-vectors of rank 1). Sometimes the term k-multivector is used in place of k-vector. I think that the intention of making it more readable is achievable, but it would be a pity to lose understandability due to the wording becoming too fuzzy in the process. —Quondum 18:14, 26 March 2014 (UTC)
replies: text: I welcome all suggestions.  However, please notice that exterior algebra generally refers to differential forms and geometric algebra is a non-traditional formulation of Clifford algebra.  These are distinct from what is a relatively simple concept of a multivector, which is the product of two vectors.  The use of the word rank for multivectors is traditional, where as the term grade arises from its structure as a graded algebra.  I purposely have left the original lead in place, so please cut and paste so it fits your preferences.  By the way, a Clifford number is a distinct object, so introducing that concept at this stage is not helping. Prof McCarthy (talk) 18:42, 26 March 2014 (UTC)

text: I might add a couple of points.  There is only one vector space that provides support for the construction of multivectors.  The resulting multivectors form additional vector spaces.  And if this is not clear, then some simple edits should take care of that.  I might add that a vector space with a linear product operator for vectors is called an algebra, which is where the terms matrix algebra and vector algebra come from.  I am not sure how this can be imprecise. Prof McCarthy (talk) 18:50, 26 March 2014 (UTC)

text: By illustrations, I assume you mean the descriptions of multivectors in R2 and in R3.   The important feature of multivectors illustrated in these two sections is their direct connection to the measurement of area and volume in all dimensions.  This is what these examples demonstrate.  So I disagree with your opinion on this point.  But, because I am making the revisions is it the case that your opinion must stand?  I have had trouble with this aspect of Wikipedia in the past.Prof McCarthy (talk) 18:58, 26 March 2014 (UTC)

text: I have tried to address the concerns with a number of revisions.  Please be patient with the repetition of the bivector an three vector calculations from the section on exterior algebra.  My goal is to introduce multivectors on projective spaces to motivate their importance as coordinates for hyperplanes, which is how they were originally formulated before being adapted to calculus through reformulation as differential forms. Prof McCarthy (talk) 21:20, 26 March 2014 (UTC)
replies: text: My opinion stands only as an opinion. I apologize for my tone.  We haven't had opinions from others, and I'm relatively uncertain on several aspects, so it's not going to help for me to interfere.  —Quondum 05:13, 27 March 2014 (UTC)
replies: text: Thank you.  I will be attentive to your concerns. Prof McCarthy (talk) 13:16, 27 March 2014 (UTC)
thread_title:  Changes not helping 

message: Substituted at 02:21, 5 May 2016 (UTC)
thread_title: Assessment comment

message: I propose to merge Blade (geometry) into Multivector. I think that the content in the Blade (geometry) article can easily be explained in the context of Multivector, and the Multivector article is of a reasonable size that the merging of Blade (geometry) will not cause any problems as far as article size is concerned. Rgdboer (talk) 00:47, 10 January 2019 (UTC)
replies: text: The term geometric algebra was used by E. Artin for matrix methods in projective geometry.<ref  E. Artin, Geometric algebra, Interscience Publ., 1957 /ref>
The book is Geometric Algebra and has nothing to do with multivectors. Sweetening with mention of Artin is deceptive. — Rgdboer (talk) 02:46, 13 January 2019 (UTC)
thread_title:  Merger proposal