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Columns of Rref - Linear Algebra - Quiz Solution, Exercises of Linear Algebra

This is the Quiz Solution of Linear Algebra. Mainly includes points are Explicit Conditionsm, Expansion Across, Equilibrium Prices, Equation, Elementary, Elementary Row etc. Key important points of tags are: Columns of Rref, Columns, Matrix Product, Mean, Rst Question, Definition, Vectors, Subspace, Basis, Vector

Typology: Exercises

2012/2013

Uploaded on 02/27/2013

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Math. 205B Quiz 05 page 1 10/30/2009 Name J.lJjt;/tl [o/k>
Here are some facts:
[
-4 -8 -16
6 19 31
Let A =
.
5 16 26
3 10 16
b~ [~lu~ [~lv~
8 -10
16 26
14 20
10 17
3
4
-73
2
3
-14
]
20
7 ,and let
26 -13
24
4
3
-255
and w =
Let the oolunm.~of A be CI.. .C6, and let the oolumns ofrref(A) be kI.. .kt;. Let R =rref(A).
[
102 -10 0 7 08 -7 -4
]
0 1 1 4 0 -8 0-25/9 8/3 10/9
Fact 1. The rref of [A I 14] is 0 0 0 0 1 5 02/9 -1/3 1/9
0 0 0 00 0 1 12 -10 -6
I1n~~@(wntra--)rf.- ge..~
~IJ
2'(
'1
,- :I
~2>
>"
a../Il. Cor"'rj .1 J.~ J~ IL bftSIJV(clVfJtf."v
lis /"'{1I~,t;t?1(;. ~= i~-to:S~+S ~
7 _20
]
(; I->-
{
-z.
]
-1
[
/~
]
~v.- -\ 11.- ~
-8 -15 IIZ- -~ 3f/t,A.
~ 1~ 't:..~ ! 1 ~~iIjf)
Fact 3. Finally, the rref of
r
-4 -8 -16 8 -10 -14 30
]
6 19 31 16 26 20 37
5 16 26 14 20 7 0
3 10 16 10 17 26 79
is
[
1 0 2 -10 0
0 1 1 4 0
000 01
000 00
J
i
-t
aJ. ~::- 8
The fustquestion;"
lure, ~-~Jf
1.What does it mean (ie, what's the definition) for a set Bof vectors {VI... Vp}to be a basis of a subspace H of a vector
space.
TL. U: 1) y)IIV,t...
(!) S:fftfI H
(Y J;. ~ (1. Jet
-13 3
[-4 -8 -16
8
-10 -14]
24 4
[0 W]
6 19 31 16 26 20 4 -7 is 037
Fact 2. Thematrixproduct 5 16 26 14 20 7 3 3 0 0
3 10 16 10 17 26 -25 2 079
5 3
pf2

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Math. 205B Quiz 05 page 1 10/30/

Name J.lJjt;/tl [o/k>

Here are some facts:

[

Let A = .

b~ [~l u~ [~l v~

]

7 ,and let 26

  • 24 4 3
  • 5

and w =

Let the oolunm.~of A be CI.. .C6, and let the oolumns ofrref(A) be kI.. .kt;. Let R =rref(A).

[

]

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 _

]

(;

I ->-

{

-z.

]

[

/~

]

~v.- -\ 11.- ~ -8 -15 I I Z - -~ 3 f/t,A.

~ 1~ 't:..~! 1 ~ ~ iIjf)

Fact 3. Finally, the rref of

r

]

is

[

J

i

-t aJ. ~::- 8

The fustquestion;"lure, ~ - ~ J f

  1. 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

  1. 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: [

]

. c/()(!$ 0 = JO ~ /2 'Sf - lOot> .

-'. .

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

  • -'2.

] [

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)