Talk:Geometric algebra: Difference between revisions
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::::::::::Put differently, I do not know whether the presence of a <math>\mathcal{V}</math> within a clifford algebra excludes the existence of a different <math>\mathcal{V'}</math> from the algebra. Maybe it does, and maybe the statement "It is a known fact that if the dimension of <math>\mathcal{V}</math> is <math>n</math>, then the dimension of the algebra is <math>2^n</math>." hints at the axioms or constraints or concepts which require that. Or maybe not. I've not studied the subject enough to say for sure. --[[User:Rdm|Rdm]] ([[User talk:Rdm|talk]]) 20:15, 19 October 2015 (UTC)
:::::::::::Notice that I did not mention bivectors in the task description, only in this discussion page. The task description hints that there are multivectors that are not vectors. Without any knowledge of the subject, that means that the reader can not assume anything about i, j and k because he does not know if the geometric product of two vectors is a vector or not. He should consider them as multivectors and as such the scalar product formula can not ''a priori'' be applied to them. I would also like to point at that you were the one who wanted to discuss the orthogonality of i, j and k (for which the concept does not apply), but there is no need to consider that for solving this task anyway.--[[User:Grondilu|Grondilu]] ([[User talk:Grondilu|talk]]) 20:39, 19 October 2015 (UTC)
::::::::::::But "geometric product" is not a part of the definition of "vector". You do not need a "geometric product" for a vector to be a vector, and in fact in the general case vectors exist without any geometric product being defined. Meanwhile, the axioms of the of the algebra require that a scalar product can be applied to i, j and k. So the reader should indeed be able to assume that the scalar product formula can be applied to them.
::::::::::::Still, yes, you are using a particular definition of "orthogonality" - it's not what I would have expected, and I can easily imagine other definitions which might serve in other contexts. So that means that that definition of orthogonality needs to be in the task description. --[[User:Rdm|Rdm]] ([[User talk:Rdm|talk]]) 20:48, 19 October 2015 (UTC)
== "Orthonormal basis" ==
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