Talk:System of linear equations

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Can homogeneous and inhomogeneous also describe systems of linear algebraic equations.

Hi- I am working on this Wikiversity lesson that involves linear equations of the form

A_{ij}x_{j}=b_{i}

(summation over repeated indexes is implied)

I always thought this was called an "inhomogeneous" set of equations. I also saw the same terminology used in this MIT pdf file:

http://ocw.mit.edu/courses/mathematics/18-02sc-multivariable-calculus-fall-2010/1.-vectors-and-matrices/part-b-matrices-and-systems-of-equations/session-14-solutions-to-square-systems/MIT18_02SC_MNotes_m3.pdf

I also noticed that you mentioned homogeneous and inhomogeneous back in 2007, on this talk page. If nobody complains, I will work those words into one of the early sections. --guyvan52 (talk) 22:10, 27 November 2014 (UTC)[reply]

The description of homogeneous and inhomogeneous (called here "nonhomogeneous") systems is the object of section "Homogeneous systems". I agree that this concept is important and that this section is misplaced. However, this classification is really important only for systems that have infinitely many solutions, for which it allows a finite description of all the solutions. Therefore, one may think of merging this section with section "Describing the solution". This should also be mentioned in section "Solution set" for the case of infinitely many solutions. IMO, the issue may be solved without too much work by simply linking to section "Homogeneous systems" at these two places and possibly in the lead. D.Lazard (talk) 23:10, 27 November 2014 (UTC)[reply]

Thanks, I made the link as per your suggestion right here in the Wikiversity article. The words you chose suggest that you think that perhaps this Wikipedia article introduces the concept a bit late, and I agree. But no article can be all things to all people, and I have participated in enough such discussions to know that compromises must be made. The important thing is that this Wikipedia article does explain the concept, and in a way that permits a sister-link from Wikiveristy. — Preceding unsigned comment added by Guy vandegrift (talk • contribs) 02:17, 28 November 2014 (UTC)[reply]

Assessment comment

The comment(s) below were originally left at Talk:System of linear equations/Comments, and are posted here for posterity. Following several discussions in past years, these subpages are now deprecated. The comments may be irrelevant or outdated; if so, please feel free to remove this section.

Section on "other methods" is incomplete (probably needs to be split of in the end). Perhaps some more references. History. -- Jitse Niesen (talk) 07:57, 29 August 2007 (UTC)[reply]

Last edited at 07:57, 29 August 2007 (UTC). Substituted at 02:37, 5 May 2016 (UTC)

Type of a system - general solution

What is the general solution of a system of n equations one being a sum of n components c(i) or b(i) and n-1 ratios of the terms in the sum?--82.137.15.233 (talk) 19:45, 12 June 2016 (UTC)[reply]

For instance a system of 3 equations of the type:

$b_{1}+b_{2}+b_{3}=b,{\frac {b_{1}}{b_{2}}}=r_{1},{\frac {b_{2}}{b_{3}}}=r_{2},$

What is its solution and how it differs when for instance the last ratio is inversed b3/b2=r2?--82.137.15.233 (talk) 20:25, 12 June 2016 (UTC)[reply]

What form has the solution when considering 4 equations with 4 unknowns of the same type? Similar question for the case with n equations and unknowns.--82.137.15.233 (talk) 20:25, 12 June 2016 (UTC)[reply]

The solution of the first type of a system has the form: $b_{1}+b_{2}+b_{3}=b,b_{1}=b_{2}r_{1},b_{2}=b_{3}r_{2},b_{1}=b_{3}r_{2}r_{1}$

so

$b_{3}r_{2}r_{1}+b_{3}r_{2}+b3=b,b_{3}=b{\frac {1}{r_{2}r_{1}+r_{2}+1}},b_{2}=b{\frac {r_{2}}{r_{2}r_{1}+r_{2}+1}},b_{1}=b{\frac {r_{2}r_{1}}{r_{2}r_{1}+r_{2}+1}},$ --82.137.11.106 (talk) 22:21, 13 June 2016 (UTC)[reply]

When ${\frac {b_{3}}{b_{2}}}=r_{2}$ the solution is

$b_{2}r_{1}+b_{2}+b_{2}r_{2}=b,b_{2}=b{\frac {1}{r_{1}+1+r_{2}}},b_{1}=b{\frac {r_{1}}{r_{1}+1+r_{2}}},b_{3}=b{\frac {r_{2}}{r_{1}+1+r_{2}}},$ --82.137.8.160 (talk) 13:36, 15 June 2016 (UTC)[reply]

Link to Refdesk Math archives analysis: Wikipedia:Reference_desk/Archives/Mathematics/2016_June_17#System_of_linear_equations_with_a_sum_and_ratios--82.137.8.23 (talk) 14:17, 22 September 2016 (UTC)[reply]

"Usual" , "common", "typical"

Mathematical problems are not like the weather, where it makes sense to speak of common or uncommon events. This kind of language is used to reassure students about what they are likely or unlikely to encounter during examinations. (In the universe of past exam questions for the course at hand it makes perfect sense to talk of typical tasks!) But this is an example of a mindset where we adjust our language to the weakest students at the risk of confusing or alienating the strong ones. In other words, a manifestation of the race to the bottom. Far more dignified to use words such as "generic" etc.88.111.224.129 (talk) 09:05, 4 January 2018 (UTC)[reply]

Please be specific, and, when a word is incorrectly used, replace it yourself by another one. Here "common" is used as for "common solution of several equations", and this is a standard meaning for which I do not know any equivalent term that is "commonly used". "Commonly" is used once and could be replaced by "almost always". This would not be useful, as people would argue about the meaning of "almost". "Typically" is used once, for a most used method; this is the standard meaning.

In fact, your main concern seems to be the section "General behavior", where "usually" and "in general" are used in the sense of "generically". "Generically" is a technical term that is too technical for being used here. However, I agree for replacing "usually" with "in general", and adding an explanation of the meaning of "in general." D.Lazard (talk) 11:05, 4 January 2018 (UTC)[reply]

I find the new wording quite bizarre. There is massive overuse of the term "in general", and it is awkward terminology regardless. "General" is reserved by mathematicians for talking about a general (all-encompassing) case, not merely a usual case. Mathematicians use "expected" or "non-degenerate" to talk about what usually happens, and "degenerate case" to talk about the exceptions. I strongly suggest using conventional terminology. — Preceding unsigned comment added by 2.101.173.147 (talk) 12:55, 1 June 2019 (UTC)[reply]

The article specifies "Here, in general means that a different behavior may occur for specific values of the coefficients of the equations". This meaning is standard in mathematics and has been introduced during 19th century. This is a conventional terminology. There is no conventional equivalent, except, may be "generically", which is certainly less understood. "Expected" cannot be used, as it suppose that a probability law is defined on systems. "Non-degenerate" would be using a circular definition, as a degenerate case is a case that has a different property than the general case. "Almost always" also suppose a probability on the set of all systems. It is your right to think that the standard terminology is awkward, but you cannot talk of "massive overuse" for a technical term that is standard and clearly defined. D.Lazard (talk) 16:11, 1 June 2019 (UTC)[reply]

To me, this seems too vague to be meaningful without a constraint on what is meant by `specific values'. As defined, it is technically true that the solution to a system of linear equations is a dog-earred copy of "War and Peace", except for specific values of the coefficients (i.e. all choices of specific values).

I get that `in general' is trying to offer meaningful intuition. Each of the `in general' statements is valid precisely when the matrix of coefficients has the largest possible rank, and the matrices with smaller rank lie on a lower dimensional subvariety in the space of all matrices (of a fixed size). Therefore, if one were to pick a matrix `at random', it would almost certainly have maximal rank; similarly, a `random' system of linear equations almost certainly has the number of solutions described.

However, there are several obstacles to making this precise and accessible to the target audience. The obvious one is talking about `varieties' and `dimension', but a more technical issue is that there is no uniform probability density on the space of matrices, so `at random' is ambiguous.

I don't really know what the best answer is here, but I think the wording currently is misleading. In these contexts, `general' means that it covers all cases (ie the general solution); whereas `generic' would be closer to the intended meaning. 129.15.65.230 (talk) 14:57, 4 October 2023 (UTC)[reply]