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I do not understand the section “In constructive mathematics”. The algorithms consists of steps involving algebraic operations which make sense in any unital ring as long as corresponding expressions are invertible. I mean, maybe we need to formulate the condition of the steps as invertibility of certain expressions in the field, instead of being non-zero, but what else can get wrong ?
I added a bit to the entry.
In fact, if the system has a different number of equations and unknowns we should not talk about upper triangular matrices but rather row echelon forms, wikipedia. At least this is the usual terminology, I do not see how to rewrite without much adding (what I can not do now).
The paragraph claiming that Gaussian elimination does not work over the real numbers was added by “Anonymous” in revision 3.
I am deleting it hereby. If anyone knows what was meant here, let’s add it back in with proper explanation.
An application of Gauss elimination to a particular system yields the Gram-Schmidt orthogonalization:
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