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18.06ProblemSet1SolutionsTotal: 100points

Section 1.2. Problem 23: Thefigureshowsthatcos() =v1/vandsin() =v2/v. Similarly cos() is and sin() is . The angle is .Substituteintothetrigonometryformulacos()cos()+sin() sin()forcos()tofindcos() =v w/vw.Solution (4points)

Firstblank: w1/w. Secondblank: w2/w. Substitutingintothetrigonometryformulayields

cos() = (w1/w)(v1/v) + (w2/w)(v2/v) =v w/vw.

Section 1.2. Problem 28: Canthreevectors inthexy planehaveu v

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combinationof leftsidesthatgiveszero. Whatcombinationofb1, b2, b3, b4, b5 mustbezero?Solution (12points)

The5by5centereddifferencematrixis

0 1 0 0 0C=

1 0 1 0 00 1 0 1 00 00 0 1 0 10

.1 0

ThefiveequationsCx=barex2 =b1, x1 +x3 =b2, x2 +x4 =b3, x3 +x5 =b4, x4 =b5.

Observe that the sum of the first, third, and fifth equations is zero. Similarly,b1 +b3 +b5 =0.Section 2.1. Problem 29: Startwiththevectoru0 =(1,0). Multiplyagainandagain by the same Markov matrix A= [.8 .3;.2 .7]. The next three vectors areu1, u2, u3:

.8 .3 1 .8u1 = = u2 =Au1 = u3 =Au2 = .

.2 .7 0 .2Whatpropertydoyounoticeforallfourvectorsu0,u1,u2,u3.Solution (4points)

Computing,weget.7 .65

u2 = u3 = ..3 .35Allfourvectorshavecomponentsthatsumtoone.

Section 2.1. Problem 30: ContinueProblem29fromu0 =(1,0)tou7,andalsofromv0 =(0,1)tov7. Whatdoyounoticeaboutu7 andv7? HerearetwoMATLABcodes,withwhileandfor. Theyplotu0 tou7 andv0 tov7.Theusandthevsareapproachingasteadystatevectors. GuessthatvectorandcheckthatAs=s. Ifyoustartwiths,thenyoustaywiths.

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Inthisgraph,weseethatthesequencev1, v2, . . . , v7 isapproaching(.6, .4).

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Fromthegraphs,weguessthats= (.6, .4) isasteadystatevector. Weverifythiswiththecomputation

.8 .3 .6 .6As= = ..2 .7 .4 .4

Section 2.2. Problem 20: Three planes can fail to have an intersection point,even ifnoplanesareparallel. Thesystem issingular ifrow3ofA isaof the first two rows. Find a third equation that cant be solved together withx+y+z=0andx2yz=1.Solution (4points)

Thesystemissingularifrow3ofAisalinear combinationofthefirsttworows.Therearemanypossiblechoicesofathirdequationthatcannotbesolvedtogetherwiththeonesgiven. Anexampleis2x+ 5y+ 4z=1. Notethatthelefthandsideofthethirdequationisthethreetimesthelefthandsideofthtefirstminusthelefthandsideofthesecond. However,therighthandsidedoesnotsatisfythisrelation.Section 2.2. Problem 32: Start with 100 equations Ax = 0 for 100 unknownsx= (x1, . . . , x100). Supposeeliminationreducesthe100thequationto0=0,sothesystemissingular.(a)Eliminationtakes linearcombinationsoftherows. Sothissingularsystemhasthesingularproperty: Somelinearcombinationofthe100rowsis .(b)SingularsystemsAx=0haveinfinitelymanysolutions. Thismeansthatsomelinearcombinationofthe100columnsis .(c)Inventa100by100singularmatrixwithnozeroentries.(d) For your matrix, describe in words the row picture and the column picture ofAx=0. Notnecessarytodraw100-dimensionalspace.Solution (12points)

(a)Zero. (b)Zero. (c)Therearemanypossibleanswers. For instance,thematrixfor which every row is (1 2 3 100). (d) The row picture is 100 copies of the hyperplanein100-spacedefinedbytheequation

x1 + 2x2 + 3x3 + +100x100 = 0. The column picture is the 100 vectors proportional to (1 1 1 1) of lengths 10,20, . . . ,1000.

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18.06 Linear Algebra

Spring 2010

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