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Math. 205B Quiz 05 page 1 10/30/
Name J.lJjt;/tl [o/k>
Here are some facts:
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Let A = .
b~ [~l u~ [~l v~
]
7 ,and let 26
and w =
Let the oolunm.~of A be CI.. .C6, and let the oolumns ofrref(A) be kI.. .kt;. Let R =rref(A).
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Fact 1. The rref of [A I 14] is 0 0 0 0 1 5 0 2/9 -1/3 1/ 0 0 0 0 0 0 1 12 -10 -
I1n~~@(wntra--) rf.- ge..~
~IJ 2'(
,- :I
a../Il. Cor"'rj .1 J.~ J~ IL bftSIJV(clVfJtf."v
lis /"'{1I~,t;t?1(;. ~= i ~ -to:S~ + S ~
7 _
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I ->-
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/~
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~v.- -\ 11.- ~ -8 -15 I I Z - -~ 3 f/t,A.
~ 1~ 't:..~! 1 ~ ~ iIjf)
Fact 3. Finally, the rref of
r
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is
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J
i
-t aJ. ~::- 8
The fustquestion;"lure, ~ - ~ J f
- What does it mean (ie, what's the definition) for a set B of vectors {VI... Vp}to be a basis of a subspace H of a vector space. TL. U: 1) y)IIV,t...
(!) S:f ft fI H (Y J;. ~ (1. Jet
[-4 -8 -
-10 -14]
[0 W]
is
Fact 2. Thematrixproduct (^5 16 26 14 20 7 3 3) 0 0
Math 205B Quiz 05 page 2. 10/30/2009 Name fv11tdtl jO/k
- Let A, b, U, v, and w be as on page one of this Quiz. (2A) What conditions, if any, do the entries b1,o0. b4 of b have to satisfy in order for b to be in Col(A)?
~. fi,,,t 1.) ~ ~ 0 = b, +/2L2. -10101 -&h'
(2B) Verify that U is in Col(A) according to the conditions in (2A).
ri: [
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. c/()(!$ 0 = JO ~ /2 'Sf - lOot> .
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1 (
'1-1° , = 30 + ttlf'1 - 'f14 : 'flY-lfl'1 = c> / (2C) Is k1 in CoI(A)'? Explain. ,
K; '" Ul Ii nd /" (,/(1) /,jc l ~Ut~t fAn;~I!. (iY'J.~ j)".t ;'(zll)
(2D) Are ~umn vectors of A,in ,the oolumn
.
space of R'l Explain. ,1. ~ I
p
. 0!!) i /l)Jj Os //1 !<. ft/h (If ftvt /I") L,C. j ~ t~lhrl ~ /\
I"" IW( i. tr", r: I J ~ AI~)I€ Ill. ~&I>1A ~ I ;4 1M", ~ Ir", j .£J- I oj .£vrf( ~Htl1"J a.< Ih/? ?fN> (2E) Which oolumn vectors of A form a baS1Sof CaI(A)'? (write your answer in terms of the Ci'S) - u. -> ~ ~
C1, Cz., C> (2F) Express U as a linearcombination (LCj ofthe basis vectors in (2E). (write your answer in terms of the cis, eg "3cz + 4cs")
hvt j~$ ~ 2 ~ f)..~ ~
- Oc, - /{;Cz.. -I-'/ c~ ::::.u..
(2G) What LC is 3Cl + 4cz - 7Ca+3c4+ 2cs + 3C6? and how did you find it? (eg, "I used fact 7" or "I used my calculator
tooompuw.."~;m;t::~ ~Tfrl[!fi~~.
(2H) What LC is -13cl + 24cz + 4ca + 3C4+ -25cs + 5C6?and how did you find it?
F~~ Z ~,~ r:l1= HJ (21) Does the result of (2H) say that w is or is not in Nul(A)'? (^) J)~ 2H skV5; Il~ ~ D, ~t: L W f fAIl/'/~).
(2J) Find a basis for Nul(A). tL (t)..Jredv~ I~ fui 1 v.; /111 ~ : ~ }C :; -2)( +/0.,.'1 ~ 7><,
1
] [
IO
J [
~
tr,~ s '1.'7.:: - x: - 'i ~'1 .. 8'x, N 1
/ ~
. Jt L( ~ j
-: - ~ ~ 4 ~ L )( eo; fr<.e. Sb IJ [A II).. () I 0' f .. 1 0 0 - >' 3
oj. t'
x'1 00 t#<. () , 0, ".', ~VlO~ 1"fW'.
X.r= - >x, .~, Ii Ioc.c,iI ;1/v/fA),
x~ k t~L ..' : (I~~ fJ<k ~ I) (2K) Express w as a LC of the basis vectors from (~~)i' , !:::!-O£(fYI A L~ set!;
~ IL I if r?- (;""t. b»j4~ /we)
