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Linear Combination and Matrix Equations in Vector Space, Exercises of Linear Algebra

Nalanda Open UniversityLinear Algebra

The concept of linear combinations in a vector space through examples and matrix equations. Students will learn how to determine if a vector is in the span of given vectors, express all solutions of a matrix equation in parametric vector form, and find nontrivial solutions of the homogeneous equation. Matrices and their corresponding reduced row echelon forms.

Typology: Exercises

2012/2013

Uploaded on 02/27/2013

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bg1
Math 205B QuJz 02 page 1 09/25/2009 Name wq,W .rJukns.
1. Let b and v}, V2, ..., Vn be vectors in Rm.. Complete the following sentence so that it gives the definition of linear
combination:. "\Ve sa,yb is a linear combination of the the vectors VI, V2, ..., Vn if and only if..."
,.. .;krt IJIiJjtscakrs; 0/,,.-) 01."in II( J~ k. 01,~r'" +0(...VI\ :::: ~.
~ --'
2.Wa,~ U3la,~ [J9l~~ m..~ [JJ ~,ktb~ [!] andc~ [!].
2A. Isb in the span of {at, a2, a;3, at}? Explain your answ€l". Show any matrices and corresponding rref's you use.
~Mj(rlf4J. (M/'r;x ~rffAftJtIdJ.J fi,~~K/~ i,
[
C, I10 I'][I30'1/j
~ 3 0'1 gIf> rrf ~0 0 I Zf .
-3 -'j Lj -~ 2CJ ' () () 0 0
.f . . I II. dJ ~.11!/jv'~ $rr~,( hy Ii IJq mAtfixh'
S!fJC< 1M; (H1.;/flX~VS ~Ji{J:m.' " I~ ..-10 =1.
)
, € /- I'f) /t~-liIu- I'/'t!f ,nAIr/x IY/,e:se'lfi IN ~v~ Ox,+'" )('1
InCf}{I".{ Lent(ii/leI. If,J../'U 14,'..
JQ ~!Jot in -tht ~Ct~ t!~.I ~,If) ~} .
2B. Let Abe the matrix whose columns are aI, a2, a3 and~. Express all solutions of Ax =c in parametric vector
form, that is, as P + Vh'where p is a particluar solution of Ax = c and Vh is all solutions of the correspondinghomogeneous
.equation Ax = O. Show any relevant matrices used in your work. (ci,..c/,.fM\i.J ...:>
,~ ~4j"'~.vf1;'( IY/,.,fr,'..ttK)'" I! -;\. ~ Vh/
I10 \2<;l -t ('. (}, ~ ~;fJ! 7/p '. -'" ", \
]
<
:~. ~, : 'i (
)(. g-sx -lfx ;' g .:.f -3 - \
.~X~«-t; '. / 0'
~~i{/p~,1iA.mtfj svlvhh>-I')iV(1\ 0x: [X:J
=[IO X7..- zx..,
]:r~]+:1[0]+'II, [." ;'
. .-) X..,)c., J' o..} '.. 0J/
.~X"L ~~~ t:f~' free: ' ,~,..'
2C. Use your work in (2B) to find two nontrivial solutions 81 and S2 of Ax =O. CIRC~E yom an,swers. . . dj
~uitkr -ki- ~ 9~JAll (vlvfW..tjf)t<:: -0 eyv,) dof£ recti, scl~ wefIt ~
to G~ vdJI!>fer X'lo ~)(,., IVr R/)Ct:t7~I .J-..I,u Xl.~2.awl )<,., =: 1. ;
Vh bJt)(s; 2
[
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... '[,O
~
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'" [-o
j'::' I-/~
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() -2. /) -z -z .M~et:W' Nt: ;t!ot /cJpiu,JJ,. fc:i,Ji;.;
I 0 I- ./
IiIb :::J':> /1/)
17>< C.../
2D. Now let Tbe the matrix whose cohunns are aI, a3 and ~(so Tlooks like A. if you .ake out A's second column).
\Vhat is Vianow? That is, what are the solutions of Tx = 07
(W/UIt!!It CtdtJtta.t, "AI'I~h b" ", )<.l3x")
~J [TI;] = f-~ ~
'11:1 )1,& ~'"
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I
P
J[
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I
O
J
'1-,= - '1)<.., ~
'f () t'V 0I z 0~ )(t = - 2)(... 9' "'n=
-'1 () (J 0 0 () , I'
X, c,.1'~e.
(~ X' nJ)
x,PD

Partial preview of the text

Download Linear Combination and Matrix Equations in Vector Space and more Exercises Linear Algebra in PDF only on Docsity!

Math 205B QuJz 02 page 1 09/25/2009 Name wq,W .rJukns.

  1. Let b and v}, V2, ..., Vn be vectors in Rm.. Complete the following sentence so that it gives the definition of linear

combination:. "\Ve sa,y b is a linear combination of the the vectors VI, V2, ..., Vn if and only if..."

,.. .;krt IJIiJjt scakr s ; 0/,,.-) 01." in II( J~ k. 01,~ r'" + 0(... VI\ :::: ~.

~ --'

2. Wa,~ U3la,~ [J9l~~ m..~ [JJ ~,ktb~ [!] andc~ [!].

2A. Isb in the span of {at, a2, a;3, at}? Explain your answ€l". Show any matrices and corresponding rref's you use.

~ Mj(rlf4J. (M/'r;x ~rffAftJtIdJ.J fi, ~ ~K /~ i,

[ C, I 10 I' ] [^

I 3 0 '

~ 3 0 '1 g If> rrf ~ 0 0 I Z / f. j

-3 -'j Lj -~ 2CJ ' () () 0 0 .f.. I II. dJ ~

. 11!/jv'~^ $^ rr~,(^ hy^ Ii^ IJq mAtfixh'

S!fJC< 1M; (H1.;/flX ~ VS ~ Ji{J: m .' " I~.. -10 =1.

, € /- I ' f) / t ~ -liIu- I'/'t!f ,nAIr/x IY/,e:se'lfi IN ~v~ Ox, + '" )('

InCf}{I ".{ Len t (ii/leI. If,J.. /'U 14 , '..

JQ ~ !Jot in -tht ~Ct~ t! ~.I ~, If) ~}.

2B. Let A be the matrix whose columns are aI, a2, a3 and~. Express all solutions of Ax = c in parametric vector

form, that is, as P + Vh'where p is a particluar solution of Ax = c and Vh is all solutionsof the corresponding homogeneous

. equation Ax = O. Show any relevant matrices used in your work. (ci,..c/,.f M \i.J ...:>

,~ ~ 4j"'~.vf1;'( IY/,.,fr,'..ttK)'" I! -;. ~ Vh

I 10 \2<;l -t ('. (}, ~ ~ ; f J! 7 /p

", \

]

< :~. ~, : 'i ( )(. g-sx -lfx ;' g .:.f -3 - \

. ~ X ~ «-t; '. / 0'

~ ~ i{/p~,1iA.mtfj svlvhh>-I')iV(1\ 0 x:

[

X:

J

[

IO X7..- zx..,

]

:r~

] +: [ 0

]

+ 'II, [." ;'

. .-) X..,)c., J' o..} '.. 0 J/

. ~ X"L ~~~ t:f~' free: ' ,~,..'

2C. Use your work in (2B) to find two nontrivial solutions 81 and S2 of Ax = O. CIRC~E yom an,swers... dj

~uitkr -ki- ~ 9~J All (vlvfW..t j f)t< :: -0 eyv,) dof£ recti, scl~ wefIt ~

to G~ vdJI!>fer X'lo ~)(,., IVr R/)Ct:t7~ I .J-..I,u Xl. ~ 2. awl )<,., =: 1. ;

Vh bJt)(s; 2

[

]

... '

[

,O ~ :::-

[

-f

l. '" [

-o

j

'::'

I

-/~ ]

(

j'lOfc. flu f j 20 ~ NOT/al/Jr/(~ ,

() -2. I /) 0 (^) -zI -z .M~et:W' Nt: ;t!ot - /cJpiu, ./ JJ,. fc:i,Ji;.; I

i

Ib :::J':> /1/)

17>< C .../

2D. Now let T be the matrix whose cohunns are aI, a3 and ~ (so T looks like A. if you .ake out A's second column).

\Vhat is Via now? That is, what are the solutions of Tx = 07

(W/UIt!!It CtdtJtta.t, "AI'I~h b" ", )<.l 3 x")

~J [TI ;] = f-~ ~

'11:1 )1,& ~'" N I P J (^) [

I 0 '-{

I

O

J

'1-,= - '1)<.., ~

'f () t'V 0 I z 0 ~ )(t = - 2)(... 9' "'n= -'1 () (J 0 0 () , I' X, c,. 1'~e. (~ X' nJ)

x,PD