Solving binary linear equations - Check matrix

Put r:=x4 and s:=x5. You have x3+x4+x5=0, hence x3=−x4−x5=−r−s=r+s (the latter is because we are binary). Also, x2+x5=0, hence x2=−s=s. MathematicsStackExchangeisaquestionandanswersiteforpeoplestudyingmathatanylevelandprofessionalsinrelatedfields.Itonlytakesaminutetosignup. Signuptojointhiscommunity Anybodycanaskaquestion Anybodycananswer Thebestanswersarevotedupandrisetothetop Home Public Questions Tags Users Unanswered Teams StackOverflowforTeams –Startcollaboratingandsharingorganizationalknowledge. CreateafreeTeam WhyTeams? Teams CreatefreeTeam Teams Q&Aforwork Connectandshareknowledgewithinasinglelocationthatisstructuredandeasytosearch. LearnmoreaboutTeams Solvingbinarylinearequations-Checkmatrix AskQuestion Asked 4yearsago Modified 4yearsago Viewed 353times 0 $\begingroup$ Giventhecheckmatrix $$ B=\begin{bmatrix}1&0&0&1&0\\0&1&0&0&1\\0&0&1&1&1\end{bmatrix} $$ Ineedtosolvethebinarylinearequationto0toobtainthebasisforthelinearcode$C$,butI'mgettingthewronganswer. Theanswershouldbe: $$ C=~ $$ HowdoIsolveBproperly? linear-algebracoding-theorybinary Share Cite Follow askedAug29,2018at21:39 string_123string_123 311bronzebadge $\endgroup$ Addacomment  |  1Answer 1 Sortedby: Resettodefault Highestscore(default) Datemodified(newestfirst) Datecreated(oldestfirst) 0 $\begingroup$ Put$r:=x_4$and$s:=x_5$.Youhave$x_3+x_4+x_5=0$,hence$x_3=-x_4-x_5=-r-s=r+s$(thelatterisbecausewearebinary).Also,$x_2+x_5=0$,hence$x_2=-s=s$.Finally,$x_1+x_4=0$,i.e.,$x_1=-r=r$.Therefore,thesolutionwritesas $$ x=\begin{pmatrix}x_1\\x_2\\x_3\\x_4\\x_5\end{pmatrix}=\begin{pmatrix}r\\s\\r+s\\r\\s\end{pmatrix}=r\begin{pmatrix}1\\0\\1\\1\\0\end{pmatrix}+s\begin{pmatrix}0\\1\\1\\0\\1\end{pmatrix}. $$ Share Cite Follow answeredAug29,2018at21:52 amsmathamsmath 10.1k1212silverbadges2626bronzebadges $\endgroup$ Addacomment  |  YourAnswer ThanksforcontributingananswertoMathematicsStackExchange!Pleasebesuretoanswerthequestion.Providedetailsandshareyourresearch!Butavoid…Askingforhelp,clarification,orrespondingtootheranswers.Makingstatementsbasedonopinion;backthemupwithreferencesorpersonalexperience.UseMathJaxtoformatequations.MathJaxreference.Tolearnmore,seeourtipsonwritinggreatanswers. Draftsaved Draftdiscarded Signuporlogin SignupusingGoogle SignupusingFacebook SignupusingEmailandPassword Submit Postasaguest Name Email Required,butnevershown PostYourAnswer Discard Byclicking“PostYourAnswer”,youagreetoourtermsofservice,privacypolicyandcookiepolicy Nottheansweryou'relookingfor?Browseotherquestionstaggedlinear-algebracoding-theorybinaryoraskyourownquestion. FeaturedonMeta RecentColorContrastChangesandAccessibilityUpdates Related 0 Whatisacheckmatrixforthelinearcode$\{0,1\}$? 0 Findinggeneratormatrixforbinarylinearcodegivenparitycheckmatrix 1 TheOrthogonalProjectionMatrixSizesnotworkingout 3 Numberofcheckbitsforsingle-error-correctingbinarylinearcode 1 howtocheckamatrixrepresentationofalineartransform 0 Paritycheckmatrixforbinarylinearcode 1 obtainparitycheckmatrixindifferentways HotNetworkQuestions SIngle-directionallinkedlistinC Frenchequivalentoftheidiom'flogadeadhorse' AsaPhDstudentintheoreticalphysics,whatfractionofmytime(ifany)shouldIspendonbackgroundmaterialratherthanmyresearchproblem? Necessityofdecodinginfault-tolerantquantumcomputation Lookingforashortstoryaboutthehighestflightever PricingabonddenominatedinUSDbutissuedinEurope Eraumavezumgatomaltêstocavapianoefalavafrancês TravelonlastdayofSchengenvisa DoeskillinganapplicationkilltheSQLconnection? HowdoIgettemperatureanddiameterfromalightcurve? Whydoes"sei"appearinthissentence? Howisaphrasepatternsubdividedandmeasured? What'sawordforfakepoliteness? WhymightSiemensimplyGFCIbreakerwillreduceprotectionon3-wiredryercircuit? Whatisthisrhythmgame? Crackopenvsopen? Whyissendingtroopsdifferentfromsendingmilitaryequipment? Creatingaduotonewithonecolorasthebackgroundandonlyblackforshades Batterywithlow-voltagecut-offcomparatorcircuit Determiningtheproblemtype-VehicleRouting-Taskassignment Doesanyoneknowthetitleandauthorofthisbookwhereaboy/youngmangoesintoaroutineoperationandthesurgeonsdiscoverthathe'sarobot? Sci-fihorrorshortstoryaboutalienthatwantstobeeaten AlternativeformulafortheBernoullipmf? Isreleasingthiscompany'sdocumentsillegal? morehotquestions Questionfeed SubscribetoRSS Questionfeed TosubscribetothisRSSfeed,copyandpastethisURLintoyourRSSreader. Yourprivacy Byclicking“Acceptallcookies”,youagreeStackExchangecanstorecookiesonyourdeviceanddiscloseinformationinaccordancewithourCookiePolicy. Acceptallcookies Customizesettings