Math 205A Final Exam (75 points)

Name: SO \~<lY\S '

. Check that you have 8 questions on four pages.

. Show all your work to receive full credit for a problem.

1. (12 points) Short answers: (Show all the calculations to get the answers. No explanationsneeded. )

(a) If C is a 4 x 5 matrix, what is the largest possible rank of C? What is the smallestpossible dimension of Nul C?

{'\w, VlV\Mbo' rl- P'VtJ b ~ If . ~O lO-rt ~n k -:::-4- .('J\.u-", be ( ut ud \A.>-1'\'Aj. :::: YO',.t'k L T of;IV)N vl c .

~ =- Lr + JJ:Yf\ }J'~ C .~ JW\cJl~t ~~ N~ c ~ r .

(b) Fora 3 x 3 matrix B, det B = -1. Find det 4B.

Jet- 4 f3 ::: Lt'3JJd: 8 C nhCL- ~~ ~ CJ

==- h> If (-r) 4. ~t'\cl -the.~ CAre-

Jek 4-& ~ -( lr

(c) Find the distance between the vector i1= [ ~3 ] and the vector 11= [ ~ ].1),-1}tJJ\ CR- ==- n Cl.-V II

=1) L6J r1= JIGfO

suJJ ~th~ Y1rWS.)

-::::4-

(d) Let T : JR3 +JR2be the linear transformation given by

T(XI, X2, X3) = (Xl - X2, 2X2 - X3).

- Find a matrix A such that T(x) = Ax.

!( [~n) = l~;~J ~ X, UJ + ~L[-n-tX3~]-;:::::

[I '-I (?

]Q

Y,

]0 2- - , X7.- -~-1 O

J2- -I)DA~ G

(e) Let Pl(t) = 1,fi2(t) = 2t, jh(t) = 4t2- 2 and iJ4(t)= 8t3 -12t. Then B ={Pl, fi2,jh, iJ4}

is a basis for 1'3. Find the polynomial qin 1'3, given that [q]B~ [ -f].

C}~ ~ ~- 3fv+\P; -to'~

-:::2-(I ) - 3 ( 2 b) t L 4:b~- ~J T 0

'2 L -6 t-+ tpP--2-, L-

r ~ ~ -b r--t lrr-

2. (4 points) Suppose the columns of a 4 x 4 matrix A span JR4.Is det A = O? Explain.

~~

1:h l1.k

Jff\(,L

~~G1h \4/.> A tV T

J-D del-- A =F 0 .

~(L 60' lArv)1\S 4- II J' pq" IIZ~ I

fl ~ ftv. b t" ewe] ntW of- A .

ft-c l.r "'- s r"Ar€.. OV\.~x, ~e.~ if "'-- fivoh'>i(A) 'Q rnY'\ <A.J1 ~.

h A-. i7 M'ALA hl~

3. (12 points) Determine if the following sets are subs paces of the appropriate vector spaces. Ifa set is a subspace, find a basis and the dimension of the subspace. If a set is not a subspace,explain why.

(a) W ={

all vectors in IR3of the form

[

S - ~~ ~~

]

h.

}were r, s, tare m IR .

2r + 5t

[(J~irJ~'EI]T SrI]~&r~J

~iJ W:::c SF" 1 f-

.

\ ] ) [f.J

) r-3 J Jke<)lt. tJ Jj lA-cfu.bJrc.L

L~ }; Ls 'o{--u2>.V1 V?-' v:)

t) r;-' 0

l U0 2-'C;

J

I__~ rfL;j ~ s~ ~ ~I -3 rv ~" / .

~ a!> 0 6 -°/ VJ,v 't;v,f L).v2--,Hy V; 0=-0 hJv., c--

r"o-n-h'" ~ .JO) " ,,~~ JJ 0... IJh~ "" "" b~7/Y) rP- V;

()(\.! '72-- . ro F!I'f>- J~ S ~ -1\..0-l--"'- &,. ,',\J. JJ b ...t P/)'~jLI ~ ~, tf"\.J. - JeI- -Ko-k, sr:f)J hI.

!--{~Q ~ c;./J IS ~ W== i ~J v1--~. ~"rY\ W::::- 2- ~

(b) W = {all polynomials in 1P'2of the form P(t ) = at + bt2 h b.

,were a, are mtegers}.

Fe tJ == 2- b+ 1:-1- r..r 1/\ vJ .c-:::- 1- JJ (A s~~f .

-.3

c.. rn) -=: 2. t -+ -.i.f: 1.- /J '" 0I:-- i'r, hi t ('..c.O.4Je....3 32:- cA/"'\d J- ~~ hOt-- 1V\~~r:J .-J 3 . '(j

~ 0 H D '1\ot- vloJ~ lA"Jd .Ic.ol9.r 'YYlt4l:ifh~~ .

go W IJ "ot-- fA .1lAbSp&.U of' {~ .

-,

(c) W = {all symmetric matrices in M } (R 11 h ..

such that AT = A.) 2x2. eca t at a symmetnc matnx is a matrix A

k S'J""~'z-~),- f" HUL ~ -r,<>- ~ [~ :JtJh L~ et), (c C\~ ~ Y)\.,{Yt)b~yJ.

[b' ~J = 0.[; ~J +b[f ;] {- G[~ n~ ~ ~

It B L

A )3 I L IX.-L fY) hJ lA']j <'-~ J 'rf\ oJ,.,~ M W GJ'>

be 0'l11--k" <v.J ~ (;"eP-Y 4>m b )Y"'i;J)~ 01- ~I 8, t .

5-0 W = JfQ,) {A, 15/ c. j .H€/\rL W 14 lL -'<!kbJpCJ-.'L* H)¥2- .M 'rf -UV' '"11f 0.,..,.,W}' hie -Ie, IR4c CA-J l-<.,JLf P,f2- Sk~

0JrrrJJ,,'£: ""e-rrl'(1J 1+ ~ tv [~J, B F Iv IT] j LOOC< ~ [n -

{U],U] Ul1 /1 C'< u>,' ""J ~ M Wlr, dO f A, r'"c3 ~ 0- ~',,- Md- ,.J-t\1 tr 2-t:1-. J-v btV) (J ~ W ::::.. -LA-)~( (S . c1~ W.:::-3 .

4. (5 points) Suppose A is a symmetric n x n matrix and B . .BABT is orthogonally diagonalizable. IS any n x n matnx. Explain why

CB It- ()'F ~ ()3rl AT~T C?WJpdf-,u .rt- /-r"'~Jroo '-)

:::=- B A' I!,T Cp.."Pe.,lf.,-w A- kY\ "F "') ,

:=- B A BT (N~ A Ji'1 CL A- iJ ",-,.(0' meJ>,LrNk~J

L6 It r{l-= 6Af) . So B,4/3-r i3 4- J((''''' eh'C: ~'x .

~tAcL It-- 11 A\)J~ olIO~,Jj~d)k .

.-

5. (12 points) Define a linear transfor~ation T : ]P\ ---t ~2 by T(f!) = [~~~~]. (Recall that avector fJ in ]P\ is a polynomial of the form a + bt.)

(a) Find T(3) and T(2 - 7t).

fL~3 fLIJ=3 . So TLpJ=DJ .

'flt);:::'2 -7 f.-. t(l) ~ 2.-1= -s-' 5'0 .,c.p ~ ~~J

(b) Find a polynomial p in ]P,!such that T(f!) = [ : ] or explain why we cannot find such apolynomial.

Let-- h :=::- lA+ bl:- T(;-):::: [pLJ)]

:::::[D.-f h

J(AL W~f).f-14J

r . vr '-UJ Ctfh'P

.Jo I ~ r:: ] . J;" 6-.+b =- it . 1hVJ!'TV\"-(J Gc"S~ (,crt'.-

pUJ&lbl~, O"~ "'f).\~ /J ({=-f, 6=3 ' pH.J:;:; If31:-.~o Rt-Y (Ar,Jwif 1J fN;;1- / b~ 1. 7)Ct-J== ~ +:1- L-- .

2- 2- r 2- ~

(c) Find a polynomial that spans th~ kernel of T. (Recall that the kernel of T is the spaceof all vectors that are mapped to the zero vector under T,Le., the kernel is the null space

Of;~~+ bl;. l(p)~ ~~ [f2J~[~Jk r;;tJ=roJ~ CAt h-:::--0 !2 b~ - 0.-

-n~ ! r =- 0..-~ =- LA. [H:J ' H V) u.- ~Ih e,( of'. 7~ [fCct) (Hd .

(d) Is T one-to-one? Explain.

We.. Stu:J /r, ~k Cd ~ T~f)~O ~ )!)hy" fi[(f NW.(j

J-o\vvh~J. rhl,t-j, r]" "6 -b- e1Y1e..- to -crY)L '

[

21 1

]6. (12 points) Let A = 1 2 1 .

112

(a) Is 1 an eigenvalue of A? If so, find a basis for the eigenspace corresponding to 1. If not,explain why not.

C,tt-I)X~b '4-"L=rl " 111

\J \ I J ~

G

'

j [X,:-X?1

][.

\ \ I 0 ] ~

J I \ D

]>\, ~t..~X3- x", c.\~ =

.

XJ-

I I I () rV 0 V 0 D ~ ~ J3, I I 0 0 D 0 D j '1.3 '. 0 -

[-)

J't [

I

J?:-. X)..- \ t':3 0b I

~L /'.JjV1' Cft-l) 'i~ hM I"-WJa ""'-OC"O.ro1,Jj~. £-0 I iJ' "1'1

~VZck rrf- IT .{

-, -/ L ''ThL Se)- i:s tt'r)~ I'nJ '

)~ H lS .(:,.,.- ~~ f"-!L;=C [,} [ ? J J .M~ Sf"f\S 1J,e .1 (J (:) €-t rJp~CL .

(b) Is [ ; ] an eigenvector of A? If so, find the corresponding eigenvalue. If not, explain

ATD~u~~J[!J=LtJ~~UJ,~So [)J [J It" 11.aW A A trnJ. fi,e (frYrlA r '" if-

tAO~~ to 4- .

(c) Use your answers in parts (a) and (b) to diagonalize A, if possible. (That is, if possible,find matrices P and D such that A = PDP-1.) If not, explain why not.

r-J] [

-I[

J

~b I ,J I :J lV~ ~ ~ A- .~., hA VL.-. CL

[-

.

1 q -I I] [

I <9 0

J

HilmI--,

'.

h -e...ve..Y"-f co)Uh-I r") "

f~O J (\lOt D,r V n

() ,! '1) 0 I (',; n,tP<- ec~Vt.A1f>O OcYt- Uh. I~ .

~tL At ~ -h,~ lin. l"'cL ~~) A iQ ctt1McJRoj,L

P~

[

- r - , I

]7)-

[' D O

JI 0' / -D { D0 (( 00 It .

~

1J

7. (10 points) Hurricanes develop low pressure at their centers that generates high winds. Themaximum wind speed s (in knots) and the central pressure p of a hurricane are approximatelyrelated by the equation bo + bIp = s. We have the followingdata on four recent Atlantichurricanes in the United States.

bo+bI ['1()~) ~ I3vbof 10I CCJ2---0) -=== r) 0

66f" b, L1CD)-=:-g0

bo-f hi C~~0) =:-b0 .

(0 b'n.~ ~6) h, Me Jo \~ r., t. ~'n A;z ~ 5[A-J] tV \h ~_~l. ~rnr 0 coj' &, -/h e.- S '((!k.,., i.r 'M to-F1! IJ fu, 1-

~ ~ lNo-e. k huYlYlol e-rhS, If 4- J( ~r s f<:, t,YIJ -h-e.-

I~ r -:5,\NA.~ .:S-o\r) ./l:TA -:::= [f I r 1 J

rf 90S-[»''lo ~{o ~1o II ~~

( q'fo

1\oJ rk~] = [i~soJN~ M- Sol~ 1iL ~Y)' A1/tx:;:;-. ftT S

[

4 317 ~ 3SiJ ]r ~ D

377'> 3~'11'2.5" .3 'DO 9J tV La I~ hb ~ fS'Sb / b,==-0,8

Find boand bI so that the model bo+ bI p = s is a least-squares fit to the data. (Start byusing the given data to write a system of linear equations to determine boand bd

. -r'i30

J

-

[6

I S = J' D x:::: O

Jgo I hI6D

A -:::- r t cr t) t7I ~ 2-0

1 q b0

( 410

- rLf 377~J?- [277S- 356,1'2-5"'

Pr(f -::: rt I I[to.> ~2--0 1G D

rS-V

J---Ofg

.,

"

8. (8 points) Let il, = [ ~ ] and U, ~ [~]. Let W ~ Span{il" iJ,} and if ~ [ - ~ ] .

(a) The set {Ul, U2} is not an orthogonal set. Find two vectors in W that are orthogonalto each other and span W. (Use the vectors Ul and U2 to produce the two orthogonal

vectorsl.-- rJ]J_L V{-:::-~-::::--LD .

L-V[ ~ - - (~ )- 2..

[I

] f2-

Jy"::c p-r"~ Vl~ "'- t~~:~ ~ = I ~ ~ L~

- - - ~ -[

2-

J ['1-

1 [(:)

1L..t "2- = \.I 1-(.. - ~ - ~ ::;.- I' ." or- ~

)~

'-.- - --:- <"""" .- \-{ , ~ .

~ - Vi ~, J v,-=- t.t,.- IA = 14" - ~ >-- '-'-1V ~ y .~.2-- I --- 1 ---

i If-"']) '" F).-~ ~ J)A~Y ~~",--J,C'YVJ 4- ~ I ~ .

'4, T.i~ fo\..FSr"-"£ lAJ I ~,j =-+n [ V, I V,J .AJ --- -- -

[jf

J [O

J~. V . ~ ~ A.rt.. ~l7j~

rrl~v) VI? V1-- D ' () - 1:>. 0 ()I ~ €..A-vh0 h- ~1 .

(b) Find a vector in W that is closest to if.

Ve-cf' 'j] W iz.-,J- IJ Jnor~ {- -/0 Ii

: Ii~ ) VI f: (t.~ )V;

l:v, '\/1 VJ-°V2.->

::c =t[~J+ f [n~ Lg J tUl £V g:: 19J

w~. ~ €-. ~ IY2.- 'he.~e.- ~ Are..-

~~~ .J" ~,1 aRex-