Difference between revisions of "Talk:Addition of vector spaces"
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(Edit conflict) However, the (linear) operations are not specified here in the {{M|K}}-indexed case; the written object is just the [[Cartesian product]] of ''sets'' {{M|V_i}}, defined elsewhere (though, for now only for two sets). Here the point should be, how to turn the product (defined elsewhere) into a vector space. [[User:Boris|Boris]] ([[User talk:Boris|talk]]) 18:23, 18 March 2016 (UTC) | (Edit conflict) However, the (linear) operations are not specified here in the {{M|K}}-indexed case; the written object is just the [[Cartesian product]] of ''sets'' {{M|V_i}}, defined elsewhere (though, for now only for two sets). Here the point should be, how to turn the product (defined elsewhere) into a vector space. [[User:Boris|Boris]] ([[User talk:Boris|talk]]) 18:23, 18 March 2016 (UTC) | ||
| − | Now about ''K''-indexed external direct sum of (copies of) the scalar field ''F'': this is also known as the vector space of ''formal linear combinations'' of elements of ''K'' (with coefficients in ''F''). Each element of ''F'' corresponds canonically to a vector of this space, and these vectors are a basis. This way, an arbitrary set ''K'' (no structure needed) becomes a basis of a vector space. [[User:Boris|Boris]] ([[User talk:Boris|talk]]) 20:37, 18 March 2016 (UTC) | + | Now about ''K''-indexed external direct sum of (copies of) the scalar field ''F'': this is also known as the vector space of ''formal linear combinations'' of elements of ''K'' (with coefficients in ''F''). Each element of ''F'' corresponds canonically to a vector of this space, and these vectors are a basis. This way, an arbitrary set ''K'' (no structure needed) becomes a basis of a vector space. [https://en.wikipedia.org/wiki/Free_module#Formal_linear_combinations WP] [[User:Boris|Boris]] ([[User talk:Boris|talk]]) 20:37, 18 March 2016 (UTC) |
Latest revision as of 20:40, 18 March 2016
About the alternative form of the external direct sum: I'd say, [ilmath]f[/ilmath] and [ilmath](u_1,\cdots,u_n)[/ilmath] are already two alternative forms of the same object (or else, what the tuple is?); and if so, then this remark is not needed in this article. Just imagine that we always list all combinations of all known alternative forms of all subexpressions of a given expression... Boris (talk) 18:12, 18 March 2016 (UTC)
Oops, no, I should read more! This is a preparation to the following [ilmath]K[/ilmath]-indexed case. Boris (talk) 18:15, 18 March 2016 (UTC)
- This was put on hold until I could unify several definitions, I still haven't worked it out (see Notes:Vector space operations) it is lacking. Writing things in the main namespace is a commitment, I was premature there. I'd love some guidance on the issue. Alec (talk) 18:19, 18 March 2016 (UTC)
(Edit conflict) However, the (linear) operations are not specified here in the [ilmath]K[/ilmath]-indexed case; the written object is just the Cartesian product of sets [ilmath]V_i[/ilmath], defined elsewhere (though, for now only for two sets). Here the point should be, how to turn the product (defined elsewhere) into a vector space. Boris (talk) 18:23, 18 March 2016 (UTC)
Now about K-indexed external direct sum of (copies of) the scalar field F: this is also known as the vector space of formal linear combinations of elements of K (with coefficients in F). Each element of F corresponds canonically to a vector of this space, and these vectors are a basis. This way, an arbitrary set K (no structure needed) becomes a basis of a vector space. WP Boris (talk) 20:37, 18 March 2016 (UTC)
