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Unformatted text preview: Math 107: Linear Algebra and Differential Equations Practice ﬁnal exam Name: : Friday, December 11, 2009 Lecture section: 107.0 Recitation section: 107R.0 All answers must be justiﬁed. No calculator is allowed. Question 1. Find conditions on a, b, c so that n = (a, b, c) in 8&3 belongs to W = Span('u1, n2, n3), where
u, = (1,2,0), in = (—1,1,2), in = (3,0, —4)
Set V= xw+_._:yuz + 2013
a (ab, c) = mam) + 9‘“““”* H 3’0! “”49
_. (x43), 3%, 2x+ g, 21y 4?) 229‘ =3 helm» 'fb Pecke (ab'fvrn/l:
o 6 ~61 vaA
0 0 Di lie26 +3C Question 2. Find the general solutionIdf; the system ' :5’1 2 2532 + 2:03
2:5 2 2561 + 2:173
333 = 2$1 + 2562 chew my is z
del{~A .—2)\ zip:0 =,> 6X+Z)C)\:‘*)=°
2 2  /\'"Z , ‘+ m A=‘t v.= [I] ._  '0
EVAE'Z V;=Lé] ,.V3"'4l‘§
_3..j ' “I .
I _, 0 .25'
pm»): q[(e t+ (Janie 2r+c3[’]e
l Question 3. Find a basis and dimension of the subspace W of R3 where (”W {mm WW: 05 0,23%!» 5.; dual/11(3) #1933
Lﬂg‘f'bﬁu a be PM var 7h a+b+ cw owe/33M ' .. —0 NL '{DVO Mepehaﬁcm‘ VCG‘IDM
— J M2 — I mg 0mm var>2 , ow! MUM; ﬁrm “Laws gem Question 4. Solve the initial value problem m’ = Am with So. x ((L) —~ I'Ie’2t(6o§5t “301513).
 2 '5
.. ' #5 m 09190115 2
f. ~2t PC0397 e t Question 5. Let V m P(t), the vector space of real polynomials. Determine Whether or not W is a
subspace of V. Explain. (a) W consists of all polynomials with integer coefﬁcients. (‘0) W consists of all polynomials with degree 2 6 and the zero polynomial. No. 179“”): 7'36 few
£64): —'{:‘g+f5 Luv} 71;a.)..£=e) 1‘5?” (c) W consists of all polynomials with only even powers of t. ((95 (ﬁww dwell?) Question 6. (a) Find the matrix of fundamental solutions for the homogeneous system [ZiHéiHEi
:4 =4 => 9cm =(zet
x =><+g¢x+<a€t 9M6): Cater 46161: t tat
m = e
0 e k
(b) Find the general solution for the nonhomogeneous system
mt—  [31] 1
t ”t ”t { I
1 [e {'8 6’
"(Me 9*“ " [o :tyfelﬂé 9* 2/ €641
= ttctIiefj
mil 9" KG): Eefﬁﬁte‘f: tattie? , Question 7. (a) Find an invertible matrix P and a diagonal matrix D such that A=PDP1 A: 6 ‘2]
0M __ ’ [6 —1
Final (grammes ¢ agewb" °T ”< _ — =2 )i: 213. ,
A3?“ 3331 3 . J\=3, v=[§] so Wig g} M Pjz if (b) Find A"? Without directly inverting A. A4 :(PDP'DVI '' Pb P Question 8. Let S consist of the following vectors in 5R4: ‘d
k u1=(1,1,0,—1), u2=(1,2,1,3), u3=(1,1,—9,2), u4(16,—13,1,3) ..
(s) Show that 8 is orthogonal and o basis of m4. 0? gag {1154: 1“
COMM“ “I'MZ =(+2+03= (./__ 0 ice .
virus1 " “'0 WM?” ’0
M2 Hit: ' ‘0 M! “Y"? :0
WWW“; .. _ WM, 3 is who )
A'OGOVAIXg, S 23:3)0’53 ‘9” (Pl? MW (My W .. tr
Q/Wj (Mcpméém'f’ l/(a‘vrj 13m. (2/1345 /5 of» (R . (b) Find the coordinates of an arbitrary vector v = (a, 11,0, 0!) in 3%4 relative to the basis 8. C}: 3239': aiiSZi
Mg“, 3 b
+c+3ql
c ..— ALL/‘3. = L...
2 M2““‘2 (5 . (6q_(§b+C('3d
C4  L—uﬁru = (753'— ...
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 Fall '07
 Trangenstein
 Differential Equations, Linear Algebra, Algebra, Equations

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