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, f and (u1,⋯,un) 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 K-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 K-indexed case; the written object is just the Cartesian product of sets Vi, 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)
