I was helping someone work on a computing problem with bit vectors that reduced to finding a basis knowing a spanning set, and realized quickly that the Gram-Schmidt process does not work as expected in characteristic-2. Since orthogonality was not a requirement anyway I came up with an alternate approach, but it left me wondering: Is there a way to salvage Gram-Schmidt in this setting (a vector space over $\mathbb{F}_2 = \mathbb{Z}/2\mathbb{Z}$) to get a meaningful concept of orthogonal basis? (Orthonormal seems out of the question since I don't see a way to define a meaningful norm.)
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asked May 18, 2015 at 3:19
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$\begingroup$ Is the Gröbner Basis what are you looking for? $\endgroup$– VladMay 18, 2015 at 3:59
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1$\begingroup$ To simply extract a basis from a set of generators, Gauß elimination into row echelon form should suffice. $\endgroup$– Hagen von EitzenMay 18, 2015 at 4:37
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$\begingroup$ "Inner products" (=the sum of products of components) are not a reliable tool for detecting linear (in)dependece in characteristic two. It may happen that an entire subspace is orthogonal to itself. For example many interesting error-correcting codes (= linear subspaces of $\Bbb{F}_2^n$) have this property. $\endgroup$– Jyrki LahtonenMay 18, 2015 at 5:19
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$\begingroup$ @HagenvonEitzen: Yes that's essentially what I did. $\endgroup$– R.. GitHub STOP HELPING ICEMay 18, 2015 at 5:48
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$\begingroup$ @JyrkiLahtonen: Could you expand on that to make an answer? Is there any other definition that can replace inner-product orthogonality to recover useful properties, or is orthogonality just largely meaningless in characteristic 2? $\endgroup$– R.. GitHub STOP HELPING ICEMay 18, 2015 at 5:54
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