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Solved: Then H and K are subspaces of . In fact, H and K

Linear Algebra and Its Applications | 4th Edition | ISBN: 9780321385178 | Authors: David C. Lay ISBN: 9780321385178 62

Solution for problem 40E Chapter 4.2

Linear Algebra and Its Applications | 4th Edition

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Linear Algebra and Its Applications | 4th Edition | ISBN: 9780321385178 | Authors: David C. Lay

Linear Algebra and Its Applications | 4th Edition

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Problem 40E

Then H and K are subspaces of . In fact, H and K are planes in through the origin, and they intersect in a line through 0. Find a nonzero vector w that generates that line. [Hint: w can be written asTo build w, solve the equation for the unknown s. ]

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AGRY32000Exam2StudyGuide Lecture20:MendelianGenetics1 • TounderstandtheimportanceofMendel’sworkingenetics o Mendel-Austrianmonkthatworkedonpea-crossestofindouthowinheritanceworked o Hisworkwasnotinitiallyappreciatedbecauseitdisprovedtheblendingtheory,whichwas thepopularbeliefinthe19 century § Blendingtheorymeantthatparentaltraitsweremixedtogetherandnotconserved inoffspring • TounderstandtheelegantandscientificdesignofMendel’sexperiments o Usedpeaplantsbecausetheywereeasytoglowandtopollinateinacontrolledway § Selffertilizationaswellascrossfertilizationwaspossible o Peaplantsheusedwereallfrompure-breedinglines(selfpollinatedplantsthatproduced thesametraitsineverygeneration),andusedvarioustraits(phenotypes)toconduct propergeneticanalysis § Roundorwrinkledseeds § Yelloworgreenseeds § Purpleorwhiteflowers o Experiment1:RoundxwrinkledcrossàF1:allroundàF2:3round1wrinkled § OnephenotypedisappearedinF1,reappearedinF2with3:1ratio § Roundphenotypewasdominantoverwrinkled o Experiment2:RRxRràallround(1:1RR/Rr) o Experiment3:Rrxrràhalfroundhalfwrinkled(1:1Rr/rr) • TounderstandtheexperimentalobservationsthatledtoMendel’sLawofEqualSegregation o BasedonExperiments1-3,Mendeldiscoveredthefollowing: § Equalseparationofallelesoccurredduringgameteformation § Agametecontainedonlyoneformofthealleles § Atfertilization,gametefusionwasrandom • Tounderstandhowthefollowingconceptswerediscoveredandtheirmeaning: o Genotype:pairofallelescarriedbyindividuals(RR,Rr,rr) o Phenotype:appearanceofgenotypes(round,wrinkled) o Alleles:alternativeformsofagene(R,r) o Dominant:aphenotypethatappearsinF1oftwopurebreedinglines o Recessive:aphenotypethatdisappearsinF1andreappearsinF2ofheterozygouscross Lecture21:MendelianGenetics-ChromosomalBasisofSingleGeneInheritance • Tounderstandthestructuralcomponentsofchromosomesthatfunctionduringsegregation th o Chromosomesandcelldivisionwereobservedinthelate19 centuryby3otherscientists, Mendelwasright! o DNAisadoublehelix o Homologouschromosomesaretwomembersofachromosomepairinadiploidnucleus, oneisfrommomandonefromdad o Ploidy:#ofhomologoussetsofchromosomesinacell § Haploid(n):organismwithonecompletegenome § Diploid(2n):organismwithtwocompletegenomes o Partsofachromosome § Chromatid:oneoftwothatareattachedtogethertoformachromosome(get separatedduringanaphaseofmitosis) § Kinetochore:wherespindlefibersattach § Kinetochoremicrotubules:pushandpulltolineupchromosomesatmetaphase • Tounderstandthemechanicalsimilaritiesanddifferencesbetweenmitosisandmeiosis o Mitosis:somatic(bodycells)dividetoincreasetheirnumber § 1diploidcell(2n)à2diploidcells(2n) § 1haploidcell(n)à2haploidcells(n) § Stepsofmitosisinadiploid • Interphase(2n) • Prophase(4n) • Metaphase(4n) • Anaphase(4n) • Telophase(2n) • Daughtercells(2n) o Meiosis:specializeddiploidcells(germcells/meiocytes)dividetoproducesexcells(sperm andegg) § 1diploidcell(2n)à4haploidcells(n) § Stepsofmeosisinadiploid(sameasmeiosisx2butwithoutreplicationtwice) • Interphase(2n) • Prophase(4n) • Metaphase(4n) • Anaphase(4n) • Telophase(2n) • Daughtercells(2n) Followingonedaughtercell… • ProphaseII(2n) • MetaphaseII(2n) • AnaphaseII(2n) • TelophaseII(1n) • Daughtercells(1n) • Tobeabletofollowthefateofparentalallelesduringmeiosis o Membersofeachpairsofallelesinageneseparateequallywhengametesareproduced • Tounderstandhowmultipleallelescanbeidentifiedandusedingeneticexperiments o Discoveryofgenescanbedonethroughobservingsegregationratios § Flowercolor:+/+(red)xalb/alb(white)àF1:+/albà¼+/+,½+/alb,¼alb/alb • Redisdominantoverwhite § Fruitflywing:+/+(long)x+/sh(shortwing)àF1½+/+,½+/sh • Shortisdominantoverlong § Todetermineifanindividualishomozygousorheterozygous,crosswithrr • Ifallarered,thenitisRR • Ifsomeareredandsomewhite,thenitisRr Lectures22&23:SexDetermination,Sex-LinkedInheritance,andPedigreeAnalysis • Toknowdifferencesbetweensexchromosomesandautosomes o Sexchromosomes:chromosomesthatareconnectedtosexdetermination § 1fromeachparent o Autosomalchromosomes:chromosomesthatareunrelatedtosexdetermination § 22fromeachparent • TounderstandhowtheXY-basedsexdeterminationworksinhuman o SexisdeterminedbynumberofXorpresenceofY § FemaleshaveX/X(AbnormalscanhaveX/null) § MaleshaveX/Y(AbnormalscanhaveX/X/YorX/X/X/Y) • Yencodestestisdeterminingfactor(SRY)whichisadominantactivatorin malenesspathway • Tounderstandthesex-linkedinheritancethatwasfirstdiscoveredinthefruitfly–Drosophila(by Morgan) o EyecolorwascontrolledbyasingleX-linkedgene,Yhasnoinfluencebecauseitdoesnot carrygene § w+=wildtype,dominant(red) § w=mutant,recessive(white) o Cross#1:Femalew+/w+(wild-typeredfly)xMalew/w(mutantwhitefly) àF1:allredàF2:3red:1white § F1areallredbecauseonlyXwithw+exists § F2hastworedfemalesandonewhitemaleandoneredmalebecausew+andw exist o Cross#2:Femalew/w(mutantwhitefly)xMalew+/w+(wild-typeredfly) àF1:1red:1whiteàF2:2red:2white § F1hashalfandhalfsplitbecauseXwithw+andwexists § F2hashalfandhalfsplitagain,withwhitemaleandfemaleandredmaleandfemale becauseofmorewthanw+ • Tobeabletodeduceatypeofallelebasedonhumanpedigrees o 4typesofdisorders: 1. Autosomalrecessive:progenywithunaffectedparentscanhaveit,malesand femalesareequallyaffected 2. Autosomaldominant:disorderisseenineverygeneration,malesandfemalesare equallyaffected 3. X-linkedrecessive:sonsofaffectedmalesarenotaffected,alldaughtersofaffected malesarecarriers,moremalesaffected 4. X-linkeddominant:daughtersofallaffectedmalesareaffected,affectedfemales passitontohalfoftheiroffspring • Tobeabletocalculatesimpleriskfactorsfortransmittingdisease-causingallelesbasedonhuman pedigrees o Mutiplicationofprobabilitiesofeachparent(andgrandparents)=probabilityofoffspring Lecture24:IndependentAssortment • Toknowthegenotypesandphenotypesofgametesthatareproducedbyparental,F1,andF2 individualsinadihybridcross o Mendelusedround(R)andwrinkled(r)seedsalongwithyellow(Y)andgreen(y)seeds o RRyyxrrYY àF1:RrYy(roundyellow) àF2:9/16roundyellow,3/16roundgreenandwrinkledyellow,and1/16wrinkledgreen • Tounderstandthechromosomalbasisofindependentassortment(Mendel’s2 law) nd o Differentallelepairsassortindependentlyduringgameteformation • Tobeabletopredicttheexpectedphenotypicandgenotypicratiosofadihybridcrossand understandtheirprobabilityofoccurrence o For2pairsofallelesusePunnettsquare o Formorethan2,usebranchdiagram o Productrule:theprobabilityofindependenteventsoccurringtogetheristheproductof theirindividualprobabilities o Sumrule:theprobabilityofeitheroftwonaturallyexclusiveeventsisthesumoftheir probability • Tobeabletousethechi-squaretesttostatisticallyevaluatephenotypicorgenotypicratios o Chi-squaretest:determinestheprobabilityofobtainingobservedproportionsbychance underaspecifichypothesis § Sumof((observed-expected) /expected) § Df=#ofcategories–1 § P

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Chapter 4.2, Problem 40E is Solved
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Textbook: Linear Algebra and Its Applications
Edition: 4
Author: David C. Lay
ISBN: 9780321385178

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Solved: Then H and K are subspaces of . In fact, H and K