Math 205B&C 03/27/09 Quiz 07 page 1Narne~tdd. SU/II~ Sam

1. Suppooe that v, ~[~+ v, = UJ and v, =[;llet s= [~n

1a. Explain why VI .1 V2. if, .v,- ': (2. 1)-t-E '1-1) +(5' -I) = q-'1- 5" ~ 0

(1->

lb. Find a unit vector in the direction of V2- #v-z./I-::: ~t ~,t. =:.!II i.. iii V~ =

1c. Find xand ywhich make B= {v!, V2,V3} an orthogonal basis of R3. (Use go

techniques; your answer will involve a RREF).

I,.;(w.d V, .L "3 Jf'/JIS~J 3'1 - 1../)(1'sy -=0 , U'r, -"Ix +Sj -:::-3

/,It I'LfiJ v, ..L.vj j;I;,~INMf 3., + Ix -!J ::: 0)or,) X-) = -]

.[-'f S" /-1][10/-1 g]

RCtJrd~ft7l1d~ tAvjhl.wt(,( mwlr;x cO 1-I -] ,...,. {/ I-If>

1d. Use the formulas developed in class for orthogonal bases to find a2 for which s =alvl +a2v2+a3v3-

(You do not have to find al and a3-)

2c. Row(A) sdllh;'.1: f,~{(. ~()I-J{R)==tdv(R)

.cwJ 11 b~/~ IJr I<IJv (t<.)

~{[/t7C?']/~o ,o7.J,ffo f'l-]J &- ~td "ht.$I!j Ift?w/A)

.co/112: fl,fJ(fJ): Cui (IF). I<REF(lfT) ~ ~,h 1.,2,'f t:.rtf,;,fclf "." b/{!/{ to I7ft.IJ.1, '1,/"!JAII);{1A}If~

2e. Express r3 (ie, row 3) of Aas a linear combinatio~ r3 =xrl +yr2 +zr4- of the other three rows of t!!:d.t:(/M.

A- (Hint: you will be on familar gTOundif you write the vectors vertically to solve the problem; find x y f...) > ..)

J

and z. Or explain why there are no such scalars- Use good linear algebra methods) / V;r~~

/Jo "~/;

I" «Vv(1<) I

It ((ow(lI)~((p~).

~ )

)=S.v7..--:5s1Ig-51

O(L. ~ - -

V2.v... ]"2.. -f /2. 7- 1"2-

.

[

233 2

]

1 2 1 3 -

2. If A= 2 1 5 -6 then RREF(A) IS R=

2 1 1 2

following. vVritevectors hODz(jntallywhere appropriate.

23- Col(A) (mmw)

ii

1: 10 pm

[

JlIii

]

1v1i

-Ylii

[I

}

"

~-1&

wv.)": -'1

--:-0cP- ~Q

-II CV

f

l 00 1

]

0 1 0 2

0 0 1 -2 Find a basis for each of the

lo 000

2b. CoI(R) [Umun}

,'J

2d. Row(R) [[I (J{7t] LOfOL] (0 U (-,-1J

,,

j',,/V{ :

I

T

]

f~

][

(

][]

-~ ~XLi t~i~~!j'Jm../)1;7/lSldtV It,! ~

t

l(Z. 2.

][\

1 2. I . I 0 0 3. ~ .:> ~ ..:>

1 I IS.""" () 10 -'1 1-:9 '3= Sr,-'lr1. -tOr'"1

l. .1 l. -(, 00' 0

(7 00D'

(YJ,of~ (,Itl!: !lJiJ i~> (,J~ f~ )ti /~J J Clira A/tJI""

V0 .;l sd ~t!!!I L. T. .<1

L~~!Ij j;(Qw(lI) /