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Name: : u 1'", Mathematics 108A: Quiz 1
Fliday, Api‘il 9, 2010
I’iofessor J Douglas Meme Part I. TruewFalse~ Ciicie the best enswei to each of the following questions
Each question is W01 th 2 points 1. The set Z3 2 {0,1,2}, with the opeiations of addition and u'mltiplicetion
modulo thiee is a ﬁeld with ﬁnitely many elements. TRUE? FALSE 2 The set of vectms (123;, mg,e';3,a:4) e R‘l such that
32:; + 252 + 43:3 + 5.7.2; m 3, 3:1 + 2:32 — 5:33 + mi; = 7 is a linear subspace oi R4.
TRUE (EALSEU
w/
3. Suppose that V denotes the space of all continuous functions f : R —+ 1R which have a continuous fiist derivatives at every point of 1R, 8. vectoi space
over the ﬁeld 1R1 0f reai numbers. Then W = {f E V: f'(.1:) = 3j(ru),for all a: E R}.
is e .lillﬁgg';fubspece of V.
@l FALSE
4. Let V be the vecto: space of tile precedhig problem Then
W a {f E V : f’(m) = 3f(2:)+7,f01a11:c 6 R}
is a linear subspace of V,
TRUE (FALSE)
5. Let; V be the vectox space of the pieceding pioblems. Then
Wm H E V:f’(0) =7} is a lineal subspace of V. eggs) Part II. Give complete proofs 0f each of the following statements 1. a (3 points) Let: W be the set of soiutéons to the homogeneous linear system :13) +232 +353 +515“ 2 0’
1'1 +2.1”; +2213 +8.14 = 0.
:L‘1 +2352 +2353 +8734 W 0' What is the coefficient mam}: of this system? 1215" l '2
I Q, 2 ‘E‘ b (4 points) Use the elementaiy 10w opemsions to put the matrix in low
1educed (echelon fem}. in,» I, .' 7 w l
’7 Em E: E E .245 z s F 4
.2. .2»? . ;
, ”a t 325‘ ‘5 .«ﬁ >
) x ‘3 a. 5 “ I?“ i it; P (I; 1 X L. g 1 3 Hr
E r " J 5
I ;
w n f \ ’ ) ‘ i ‘
I f w. a . r
'3' r”? ,f‘ ’x
w .‘
i
2. w 1
! 1 l‘ W;
:3“ j
(3“ 1 L3.) ' I c (3 points) Note that the solution set W is a lineai subspace of IR“ Find a
basis f0] W .. . .~ .. iii: :1: ‘2 21:»,  '3 “w
:bi + 3:. 313? 4.. .4.) A,“ k") . l a .2 : 1
PS3 + 3 r"? "z E' 6"?) R33 :7 9“" a.
if] 7» (“ If? r‘ "5 “ If; :(x‘ALi
N31,  r‘ ‘E
. 1 r“
27 p. u“ 2‘ E if
M ”" ,2 E 5* l! E E '3
4  1 E i "
W: 4 i r”: ' ’9 i U ' 3 E %
, N . L 2 k, . 3 , g . A. I N '2
n (19?“ f l (“‘3 'E '1 IE \GJQJKQJ ”J“ i O J] 3 ! V
.v. o ‘I ~" 2 ' ’ ~
. j r \ > E E
.\ w \ 1 ; c :
Ant (J i : IO ...
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 Linear Algebra, Algebra

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