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 l 0 -x 0 0 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 0 I 2- ~,,-J 0 .
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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.
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.
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.