Math E-21b Spring 2018 Homework #2

Transcription

1 Math E- Sprig 08 Homework # Prolems due Thursday, Feruary 8: Sectio : y = Fid the iverse of the liear trasformatio [That is, solve for, i terms of y, y ] y = + 0 Cosider the circular face i the accompayig figure For each of the matrices A i Eercises 4 through 0, draw a sketch showig the effect of the liear trasformatio T ( ) = A o this face The cross product of two vectors i R is defied y a a a a a a = a a a Cosider a aritrary (fied) vector v i R Is the trasformatio T ( ) = v from R to R liear? If so, fid its matri i terms of the compoets of the vector v Sectio : 6 Let L e the lie i R that cosists of all scalar 7 Let L e the lie i R that cosists of all scalar multiples multiples of the vector Fid the orthogoal of the vector Fid the reflectio of the vector aout the lie L projectio of the vector oto L 0 Fid the matri of the projectio oto the lie L i R show i the accompayig figure [Note: Your aswer should e a matri] Refer to Eercise 0 Fid the matri of the reflectio aout the lie L [Note: As i the previous prolem, your aswer should e a matri] Fid matrices of the liear trasformatios from R to R give i Eercises 9, 0,, ad Some of these trasformatios have ot ee formally defied i the tet Use commo sese You may assume that all these trasformatios are liear [Note: Your aswers to each of these prolems should e a matri] 9 The orthogoal projectio oto the y-plae 0 The reflectio aout the z-plae The rotatio aout the y-ais through a agle θ, couterclockwise as viewed from the positive y-ais The reflectio aout the plae y = z

2 4 Oe of the five give matrices represets a orthogoal projectio oto a lie ad aother represets a reflectio aout a lie Idetify oth ad riefly justify your choice A = = B C = D = E = Sectio 4: Decide whether the matrices i Eercises, 4,ad 6 are ivertile If they are, fid the iverse matri Do the computatios with paper ad pecil Show all your work y = Decide whether the liear trasformatio y = is ivertile Fid the iverse y = trasformatio if it eists Do the computatios with paper ad pecil Show all your work 0 54 Let A = Usig Eercise 5 (see elow) as a guide, fid a scalar λ ad a ozero vector such that A = λ For additioal practice (ot to e tured i): Sectio : 5 Cosider the liear trasformatio T from R to R 7 with T 0 =, 0 Fid the matri A of T 0 6 T = 9, ad 0 0 T 0 = 7 7 Suppose v, v,, vm are aritrary vectors i R Cosider the trasformatio from R m to R give y T = v + v + + mvm m Is this trasformatio liear? If so, fid its matri A i terms of the vectors v, v,, vm 9 Decide whether the matri 6 9 is ivertile Fid the iverse if it eists

3 7 Cosider a liear trasformatio T from R to R Suppose that v ad w are two aritrary vectors i R ad that is a third vector whose edpoit is o the lie segmet coectig the edpoits of v ad w Is the edpoit of the vector T ( ) ecessarily o the lie segmet coectig the edpoits of T ( v ) ad T ( w )? [Hit: We ca write = v+ k( w v ), for some scalar k etwee 0 ad ] We ca summarize this eercise y sayig that a liear trasformatio maps a lie oto a lie 4 a Cosider the vector v = 4 Is the trasformatio T ( ) (the dot product) from R to R liear? If so, fid the matri of T Cosider a aritrary vector v i R Is the trasformatio T ( ) liear? If so, fid the matri of T (i terms of the compoets of v c Coversely, cosider a liear trasformatio T from R to R Show that there eists a vector v i R such that T ( ), for all i R Sectio : 4 Iterpret the followig liear trasformatio geometrically: T ( ) = The matri represets a rotatio Fid the agle of rotatio (i radias) a 7 Cosider a matri A of the form A = a, where a + = Fid two ozero perpedicular vectors v ad w such that Av = v ad Aw = w (write the etries of v ad w i terms of a ad ) Coclude that T ( ) = A represets the reflectio aout the lie L spaed y v Fid a matri for the liear trasformatios from R to R that is rotatio aout the z-ais through a agle of π, couterclockwise as viewed from the positive z-ais 4 Rotatios ad reflectios have two remarkale properties: They preserve the legth of vectors ad the agle etwee vectors (Draw figures illustratig these properties) We will show that, coversely, ay liear trasformatio T from R to R that preserves legth ad agles is either a rotatio or a reflectio (aout a lie) a Show that if T ( ) = A preserves legth ad agles, the the two colum vectors v ad w of A must e perpedicular uit vectors Write the first colum vector of A as a v = c Show that if a liear trasformatio T from R to R preserves legth ad agles, the T is either a rotatio or a reflectio (aout a lie) See Eercise 7

4 7 Cosider the matrices A through E elow A = B = 0 C = D = E = Fill i the laks i the seteces elow We are told that there is a solutio i each case Matri represets a scalig Matri represets a projectio Matri represets a shear Matri represets a reflectio Matri represets a rotatio 8 Each of the liear trasformatios i parts (a) through (e) correspods to oe (ad oly oe) of the matrices A through J Match them up a Scalig Shear c Rotatio d Projectio e Reflectio 0 0 A = F = B = G = C = H = 7 0 D = I = 0 Sectio 4: 40 Show that if a square matri A has two equal colums, the A is ot ivertile 0 E = J = Which of the followig liear trasformatios T from R to R are ivertile? Fid the iverse if it eists a Reflectio aout a plae Projectio oto a plae c Scalig y a factor of 5 [ie, T ( v) = 5v, for all vectors v] d Rotatio aout a ais 4 A square matri is called a permutatio matri if it cotais a eactly oce i each row ad i each 0 0 colum, with all other etries eig 0 Eamples are the idetity matri I ad 0 0 Are permutatio 0 0 matrices ivertile? If so, is the iverse a permutatio matri as well? 4 Cosider two ivertile matrices A ad B Is the liear trasformatio y = A( B ) ivertile? If so, what is the iverse? [Hit: Solve the equatio y = A( B) first for B ad the for ] 5 Let A = 5 i all parts of this prolem a Fid a scalar λ (lamda) such that the matri A λi fails to e ivertile There are two solutios; choose oe ad use it i parts () ad (c) For the λ you chose i part (a), fid the matri A λi ; the fid a ozero vector such that ( A λi) = 0 (This ca e doe, sice A λi fails to e ivertile) c Note that the equatio ( A λi) = 0 ca e writte as A λ = 0, or, A = λ Check that the equatio A = λ holds for your λ from part (a) ad your from part () 4

5 Etra prolems for those iterested i ecoomics (ot to e tured i): Sectio 4: 49 Iput-Output Aalysis (This eercise uilds o Eercises 0, 7, 8, ad 9) Cosider the idustries J, J,, J i a ecoomy Suppose the cosumer demad vector is, the output vector is ad the demad of the jth idustry is v j (The ith compoet a ij of v j is the demad idustry J j puts o idustry J i, per uit of output of J j ) As we have see i Eercise 8, the output just meets the aggregate demad if v+ v v + = aggregate demad This equatio ca e writte more succictly as v v v + = or A + = The matri A is called the techology matri of this ecoomy; its coefficiets a ij descrie the iter-idustry demad, which deped o the techology used i the productio process The equatio A + = descries a liear system, which we ca write i the customary form: + k+ k + Iterpret this oservatio i terms of ecoomics 5 output A= I A= ( I A) = If we wat to kow the output required to satisfy a give cosumer demad (this was our ojective i the previous eercises), we ca solve this liear system, preferaly via the augmeted matri I ecoomics, however, we ofte ask the other questios: If chages, how will chage i respose If the cosumer demad o oe idustry icreases y uit ad the cosumer demad o the other idustries remais uchaged, how will chage? If we ask questios like these, we thik of the output as a fuctio of the cosumer demad If the matri ( I A ) is ivertile, we ca epress as a fuctio (i fact, as a liear trasformatio): = ( I A) a Cosider the ecoomy of Israel i 958 (discussed i Eercise 9) Fid the techology matri A, the matri ( I A ), ad its iverse ( I A ) I the eample discussed i part (a), suppose the cosumer demad o agriculture (Idustry ) is uit ( millio pouds), ad the demads o the other two idustries are zero What output is required i this case? How does your aswer relate to the matri ( I A )? c Eplai, i terms of ecoomics, why the diagoal elemets of the matri ( I A ) you foud i part (a) must e at least d If the cosumer demad o maufacturig icreases y (from whatever it was), ad the cosumer demad o the other two idustries remais the same, how will the output have to chage? How does your aswer relate to the matri ( I A )? e Usig your aswers i parts (a) through (d) as a guide, eplai i geeral (ot just for this eample) what the colums ad the etries of the matri ( I A ) tell you, i terms of ecoomics Those who have i studied multivariale calculus may wish to cosider the partial derivatives 50 This eercise refers to Eercise 49a Cosider the etry k = a = 09 of the techology matri A Verify that the etry i the first row ad the first colum of ( I A ) (I A) - is the value of the geometric series j