Math S-b Summer 8 Homework #5 Problems due Wed, July 8: Secton 5: Gve an algebrac proof for the trangle nequalty v+ w v + w Draw a sketch [Hnt: Expand v+ w ( v+ w) ( v+ w ) hen use the Cauchy-Schwartz nequalty ( v w v w )] 6 Consder the vectors u =, u =, u 3 = n R 4 Can you fnd a vector u 4 n R 4 such that the vectors u, u, u 3, u 4 are orthonormal? If so, how many such vectors are there? 5 6 7 Fnd a bass for W, where W = span, 3 7 4 8 49 3 6 Fnd the orthogonal projecton of 49 onto the subspace of R 3 spanned by 3 and 6 49 6 Secton 5: 7 7 8 4, 8 Usng paper and pencl, perform the Gram-Schmdt process on the sequence of vectors,, and then 7 7 6 7 7 8 use your calculatons fnd the QR-factorzaton of the matrx 7 7 6 34 Fnd an orthonormal bass of the kernel of the matrx A = 3 4 Secton 53: 3 Are the rows of an orthogonal matrx A necessarly orthonormal? 4 Consder the subspace W of R 4 9 spanned by the vectors v = and v = 5 3 Fnd the matrx of the orthogonal projecton onto W 4 Let A be the matrx of an orthogonal projecton Fnd A n two ways: a Geometrcally (Consder what happens when you apply an orthogonal projecton twce) b By computaton, usng the formula gven n Fact 53 (matrx of an orthogonal projecton n terms of an orthonormal bass for a gven subspace) 44 Consder an n m matrx A Fnd dm(m(a)) + dm(ker(a )), n terms of m and n 46 Consder a QR-factorzaton M = QR Show that R = Q M
Secton 54: 4 Let A be an n m matrx Is the formula (ker ) = m( ) A A necessarly true? Explan x+ x + x3+ x4 = 5 Let V be the soluton space of the lnear system x+ x + 5x3 + 4x4 = Fnd a bass for V 6 If A s an n m matrx, s the formula m(a) = m(aa ) necessarly true? Explan 7 Consder a symmetrc n n matrx A What s the relatonshp between m(a) and ker(a)? Consder a consstent system Ax = b a Show that ths system has a soluton x n (ker A ) Hnt: An arbtrary soluton x of the system can be wrtten as x= xh + x, where x h s n ker(a) and x s n (ker A ) b Show that the system Ax = b has only one soluton n (ker A ) Hnt: If x and x are two solutons n (ker A ), thnk about x x c If x s the soluton n (ker A ) and x s another soluton of the system Ax = b, show that x < x he vector x s called the mnmal soluton of the lnear system Ax = b 6 Use the formula (m ) = ker( ) A A to prove the equaton rank(a) = rank(a ) By usng paper and pencl, fnd the least-squares soluton x of the system Ax = b, where 3 A = and 3 b = Verfy that the vector b Ax s perpendcular to the mage of A 3 3 5 Fnd the least-squares soluton x of the system Ax = b, where A = 5 3 and b = 9 Determne the 4 5 error b Ax 3 Ft a quadratc polynomal to the data ponts (, 7), (, ), (, ), (3, ), usng least squares Sketch the soluton 38 In the accompanyng table, we lst the heght h, the gender g, and the weght w of some young adults Heght h (n nches above 5 ft) Gender g ( = female, = male ) Weght w (n pounds) 8 5 6 6 6 Ft a functon of the form w= c + ch + cg to these data, usng least squares Before you do the computatons, thnk about the sgns of c and c What sgns would you expect f these data were representatve of the general populaton? Why? What s the sgn of c? What s the practcal sgnfcance of c? For addtonal practce: Secton 5: 5 Consder the vector v = 3 n R 4 Fnd a bass of the subspace of R 4 consstng of all vectors perpendcular 4 (orthogonal) to v
8 Here s an nfnte dmenson verson of Eucldean space: In the space of all nfnte sequences, consder the subspace l of square-summable sequences [e, those sequences ( x, x, ) for whch the nfnte seres x + x + converges] For x and y n l, we defne (Why does the seres xy + xy + converge?) 3 x = x + x + and xy xy xy = + + a Check that x = (,,,,, ) s n l, and fnd x Recall the formula for the geometrc seres: 4 8 6 + a + a + a + =, f < a < a 3 b Fnd the angle between (,,, ) and (,,,, ) 4 8 c Gve an example of a sequence ( x, x, ) that converges to (e, lm x = ) but does not belong to l d Let L be the subspace of l spanned by (,, 4, 8, ) Fnd the orthogonal projecton of (,,, ) onto L Note: he Hlbert space l was ntally used mostly n physcs: Werner Hesenberg s formulaton of quantum mechancs s n terms of l oday, ths space s used n many other applcatons, ncludng economcs (See, for example, the work of the economst Andreu Mas-Colell of the Unversty of Barcelona) n 8 Fnd the orthogonal projecton of onto the subspace of R 4 spanned by 9 Consder the orthonormal vectors u, u, u3, u4, u 5 n R Fnd the length of the vector x= 7u 3u + u3+ u4 u 5 Secton 5: n,, and 6, Usng paper and pencl, perform the Gram-Schmdt process on the sequence of vectors and then use your calculatons fnd the QR-factorzaton of the matrx 3 5 4 6 7 8, Usng paper and pencl, perform the Gram-Schmdt process on the sequence of vectors 5 3 4 6 your calculatons fnd the QR-factorzaton of the matrx 7 33 Fnd an orthonormal bass of the kernel of the matrx A = 3 38 Fnd the QR-factorzaton of the matrx A = 4 3 5, 4, 6 7 5 3 4 6, 7 and then use
4 Consder an nvertble n n matrx A whose columns are orthogonal, but not necessarly orthonormal What does the QR-factorzaton of A look lke? 4 Consder an upper trangular n n matrx A What does the QR-factorzaton of A look lke? Secton 53: If the n n matrces A and B are orthogonal matrces, whch of the matrces n Exercses 5 through must be orthogonal as well? 5 3A 6 B 7 AB 8 A + B 9 B B AB A 3 3 37 Is there an orthogonal transformaton from R 3 to R 3 such that 3 = and = 3? 45 For whch n m matrces A does the equaton dm(ker(a)) = dm(ker(a )) hold? Explan 47 If A = QR s a QR-factorzaton, what s the relatonshp between AA and RR? Secton 54: Consder the subspace m(a) of R 4, where A = 3 6 Fnd a bass of ker(a ), and draw a sketch llustratng the formula (m A) = ker( A ) n ths case Consder the subspace m(a) of R, where A = Fnd a bass of 3 llustratng the formula (m A) = ker( A ) n ths case ker( A ), and draw a sketch 5 Consder an m n matrx A wth ker(a) = {} Show that there exsts an n m matrx B such that BA = I n Hnt: AA s nvertble 7 Does the equaton rank(a) = rank(a A) hold for all n m matrces A? Explan 8 Does the equaton rank( AA) = rank( AA ) hold for all n m matrces A? Explan Hnt: Exercse 7 s useful 3 Ft a lnear functon of the form f() t = c + ct to the data ponts (, 3), (, 3), (, 6), usng least squares Sketch the soluton [Note: Strctly speakng, a functon of ths form s not a lnear functon n the sense that we use n ths course More properly, ths mght be called an affne functon] 37 he accompanyng table lsts several commercal arlnes, the year they were ntroduced, and the number of dsplays n the cockpt a Ft a lnear functon of the form log( d) = c + ct to the data Plane Year t Dsplays d Douglas DC-3 935 35 Lockheed Constellaton 946 46 Boeng 77 959 77 Concorde 969 33 ponts ( t,log( d )), usng least squares 4 b Use your answer n part (a) to ft an exponental functon t d = ka to the data ponts ( t, d ) c he Arbus A3 was ntroduced n 988 Based on your answer n part (b), how many dsplays do you expect n the cockpt of ths plane? (here are 93 dsplays n the cockpt of an Arbus A3) Explan
4 Consder the data n the followng table: a Mean dstance from Planet the Sun (n astronomcal unts) D Perod of revoluton (n Earth years) Mercury 387 4 Earth Jupter 5 86 Uranus 98 84 Pluto 3953 485 Use the methods dscussed n Exercse 39 to ft a power functon n of the form D = ka to these data Explan, n terms of Kepler s laws of planetary moton Explan why the constant k s so close to 4 In the accompanyng table, we lst the publc debt D of the Unted States (n bllons of dollars), n the year t (as of September 3) t 97 975 98 985 99 995 D 37 533 98 83 333 4974 a Ft a lnear functon of the form log( D) = c + ct to the data ponts ( t,log( D )), usng least squares Use the result to ft an exponental functon to the data ponts ( t, D ) b What debt does your formula n part (a) predct for the year? What about the year? c On Sept 3,, the debt was 5,674 bllon dollars What happened? 4 If A s any matrx, show that the lnear transformaton L(x) = Ax from m(a ) to m(a) s an somorphsm hs provdes yet another proof of the formula rank(a) = rank(a ) Chapter 5 rue/false Exercses 5
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