Name:

Math 205A Test 2 (50 points)

S~l ~ti <fY15'

. Check that you have 7 questions on four pages.

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

1. (10 points) Short answers: (Show all the calculations needed to get the answers. No explana-tions needed.)

(a) Suppose B is a 2 x 2 matrix and fI is an eigenvector of B corresponding to the eigenvalue-2. Draw BfI in the following figure. X rJ)- ~ - "'1-

2- I~ ~ - "-U-

u

)(1

~:::- - 1M,

[

2 1 -4

](b) Let A = 0 0 0 . Find all the eigenvalues of A.

0 1 2

A-) l-==-

[2..-A I - 4

l0 -x 00 J 2-~

cM---CA-HJ=-l2-,1.)l.k(l-t ~])-OtO

=C2-).)[~~)l2.-A)-D J'1...

=-(2--A) [-)...} .

~ U CA-~l)~ tv(,()

b't'~ 4- A O\~

A=2 0I

2- ~,,-J 0

.

-,"

(c) Let iJ= [ -~]. Fin~ a vector of length 7 in the direction of the vector iJ.

I/itllc:: ~ u. U ~ ~~f \ 6 ==J 2-S- ~5 .

vuk-r cIt--[e-,r 1 rn tLt.- ciA lk~tM If- ;Az::- I _~ll

H"\A/(3

1-=t[Jr 2f / ~

J~. t2-~/~

(d) Compute the orthogonal projection of [ -~] onto the line through [ ~ ] and the origin.

Wj=Lnl lA=[tJ

D{kcrrcJ I?~~vn J(--5 CJY'/oIt = (-i ~ \ \Z'l U.U)

r ~ :::: [-3 (] UJ - -h- 4- -:;:-s-

~'\A =[3 lpJ[!J:.::-'H[~=2~

~r~ )~ -==- -=r;- r~J

::: - J-[

3]

::::--

[-3/~

J.

t I~ r ~ 2-r:;- Lit- ~ !f -fr /5'

.

[

-1 0 0 0

]

2. (9 points) Let A = 0 2 0-40 3 -2 -3 .0 0 4-6

(a) Find a basis for Col A and then state the dimension of Col A.

PIVi<) -h fy, .t;y, +- fz. ~ U>)VIrY1 f\ j ;

. ~O y,..,D1 ~ fh r<JL ~ iJ h fir

Co1 '" "" 1'\ r "'" A- o..~ Jr>'lt.AJ~

j '\ ~~uJ-- .

B~IJ & Lvi Ih-i[~l al [1/J. eN"" ~1 It =3 .

Arv \ D0 I

0 00 D

(J 00 -~) -f. r;-0 0

(b) Is the basis you found in part (a) an orthogonal basis? Explain.

i - L -r;: \-I]

-

[0

J- r 0

JL-<-t' "'\ ~ L ~ J ~ ~ ~ I ~ :::cl .

~ . 4~:::: 0 I lt2-' ~ =- - 0 1== 0 .

.10 fJ,f. bCv:>~ ~ ~ Qf\ hor Iow/i:r.

(c) Let if ~ [ j ] . Find the coordinates of if with respect to the basis you found in PaIt(a).

-1 D 0 -30 2-- 0 b0 3 -2 3D 0 4- -G

fro G,ofJJ'~~ 4- 7

I D 0 3D / 0 00 0 ., -1,~0 0 0 0 "-

tJ. y, b- ~ ho/.)/l 'Pl~ r 6 l/:' .b J

rJ

[

-1 0 1

]3. (7 points) Let B = 0 1 0 .

0 2 -1

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

B-r I =-

[0 0 I

J U

0 1 O

J U

D I 0 0

10 z.. 0 . 0 2 V D NOD I 002. () 2° () ODD

X, r~ ~ ~ (~f I)X::=::- (5 kCY? fY)h'y, I JJc! YY\~(f

x:,.~ SO\uJpyU. kenk -I iJ CUI eJ'()~.1~~ ~ ~ .

X~ ~~J=~[~J.;)1) b""" S ~ ~ronJJcr

ecJjF~=~[nJ .

(b) Is [~] an eigenvector 01 B? II so, find the corresponding eigenvalue. II not, explainwhy not.

G [n ~ [g' I -~J ft] ~ rt1 =P A [1] ~a~ij) .

So [!J u no!- ~ne:r00~J,r-,- 4- E '

4. (8 points) Suppose A and Bare 5 x 5 matrices with det A = -10 and det (AB2) = O.

(a) Find det B.

Jek- LA [};1.-).==-Jtk- [4- g. B )-:::-Jd- CI\). of J- 8. Jel- B

I-L 0 -z: (-( OJ (JJ- g) UJ-- B)

00 ~J- 13;::;--0

(b) Is B invertible? Explain.

f)n~ oLJ- B -;;;4J/ B fs f\Ot- ; n V1!ihh Ie..'

(c) What is the largest possible rank of B? What is the smallest possible dimension of NulB? Provide explanations for your answers.

g 11 Y\Ot- fn~(},h 1(. ~ 15 fv T .

l",w i3 ~ Y\Dl-- ~vL. "-- r/>ro~ In e~G «> )"mfJ .

r0 l.t:-xau t- rem Ib IL 'r<t'" k 01- f, -= Lr .f\[v."" bif ,{ uol"",nJ 4- B ~ ,NO kef ~"'" Nw {5 -

J :::- ~. f ot,'m NtJ 13 .

~ ~'M NvJ B -;::- I .

1'\'w -'lY\cJu.{- pOJ.s(~I~Jm<2n!IOY1c+ Nu( [) 1r I .

[

-1

5. (7 points) Suppose a matrix A can be factored in the form P DP-1 where P = ~

[

-1 0 0

]

and D = 0 2 0 .0 0 -1

(a) Find all the eigenvalues of A and a basis for each eigenspace.

8~y~ Jt- f\ ~~ -II 1-U Jj -to -I ,S

g~l~ t-r ~F5fu--tt o.o~r~ d

i rlJ ) L~]~ . "

I?O-JI~ f=- a(Jrr~(L WdUrovJJCJ to 2- u

~ [IJ ) ,

1 -1

]

1 01 1

(b) If possible, find a basis for JR3consisting of eigenvectors of A. Explain clearly why thebasis you find satisfies the two conditions in the definition of a basis. If it is not possibleto find such a basis, explain why not.

f rJ T J S~CZ r eJ

~"v~ ~€- ~'r' "i

-A-\1 0 / Cu k~ H~ h CArt- eAf;fr/u.k+-

~L D>lun\l1J A- r .

51 f\ ~ fi., t rt- if 0.. JI\A? {-- In

IT I"I'UI-'\J In lLfU\d.tA '

~nLL 1h~~ u '" rjwf //1 ~"-{J ~/ -h,eJ4 {y.:ruIJJ p~1\1 1ft.HIP t.L -h-e $u. {f,,RJj5;~ if 0.. bG..GiT ~ 1/(3, Jt1.so

PI ,J,.., t "rt ~ ~ u(- It ' --r~ IN?, 11-t.. b0 {f i.{)n1 f! b

<4 60~ 44,

Mw-thhI ~ . W fKJ P2-I ~ b~.~3

€-v(J "",/umf\ / h,c rd- Ir,FJ{;j

-0

6. (4 points) Let H = { [ ab+ib~c ] a, b,c me real numbers}

Is H a subspace of ffi.4?Explain.

.

L I / 0 br:l+rb - CA- -2-IA;L '5 6/Sb ,

\\.1--5

k p0 [[-11/L]I U]JHef' Lt M /[ C( .nJ"7 CA 01- (f. .

7. (5 points) For a certain animal species, there are two lif~ stages: juvenile and adult. Fork ~ 0, let Xk be a vector in ]R2that denotes the population of the species at the end of yeark. The first entry in the vector gives the number of juveniles and the second entry gives thenumber of adults. If A is the 2 x 2 stage-matrix for the species, the population is given by theequation Xk+l = AXk. Eigenvaluesof A are 1.2 and -0.4 and the corresponding eigenvectors

are [ ~ ] and [ -~] respectively. The initial population is 100juveniles and 55 adults,i.e.,

[

100

]Xo = 55'

(a) Write the general solution for the population equation, Le., write an expression for Xk.

Let- ~1~{.2-, >2-=-0.1- / 'v; =- [fJ I V2- = LiJ- .--1\) =- S~ -tCJ-V'l- .

[8 -4 {O'D] rOfr 0 folb (» L9 r -§J

C2-'.::"- r; .~ ~ [ \ )\c. .,-

X ::::- CICAI) VI t Lz- t1'l. V:z-~ k

r.t Y; -=-- }o (\,2-)<:[l?

]f-(-5) [-0.4,) r-t J

~ tL: b LI

~:::: [0/

\h V\5

(b) Use you answer in part (a) to discuss the population growth of the species in the long

run. kA~ k~CD / (-O~Lf) ~ 0 .

So 'f n he..{o1 "'1 n / ""X,,, ~!

\~~I ~e.- p~~'vY->

fu- ~~ ')'\An .

(t. 2-) ~[~J

mc Y<.0-1 w a.!- R d-c,J (J-r~ 1'>1