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MATH 220
Name
FINAL EXAMINATION
~v\ u.. ~ c "'oS
IDecember 13, 20011
ID
Section
#___
There are ??multiple choice questions. Each problem is worth 5 points. Four possible answers
are given for each problem, only one of which is correct. When you solve a problem, note
the letter next to the answer that you wish to give and blacken the corresponding space on
the answer sheet. Mark only one choice; darken the circle completely (you should not be
able to see the letter after you have darkened the circle).
THE USE OF CALCULATORS DURING THE EXAMINATION IS FORBIDDEN.
CHECK THE EXAMINATION BOOKLET BEFORE
YOU START. THERE SHOULD BE ?? PROBLEMS
ON ?? PAGES (INCLUDING THIS ONE).
l
MATH 220
FINAL EXAMINATION
~umLiUli
1. What is
of following system of equations
+ 2X2 + X3
-2XI + X2 - 2X3
3
+ 3X2
4?
1
Xl
-Xl
@
Xl
= -1-
b)
Xl
= 1 + X2,X2 is free,x3 =-1.
c)
Xl
= -X2 + X3, and
d)
Xl
= 2X2
X3,X2
X3,
=
and
l,x3
X3
l'
is free.
X2, X3
X2) X3
-
\~
-'1 L \ -:l
~
,
_\ -0 -\ 4
"X.\-;-)'l-e:.-=
"><'2
I\J -::..
are free. -1
[ ' 2\
rJ
)l\
1
0
1'0.)
r
\)
~c\u. \::{OA
-=-
0
Q
Cl
\
0
J
Q
kIS
S
tV
are free.
o
€ q \.\ C\ -b..QV\s:
PAGE 2
N
J
1{) 1~
0
5
[~~
()Q)oI
o
a {)
IJ
-j
0
_\_x~\
?L"2.. =
A 'B \ ~
-\
~ f'<,.ll..
2. Which of the following matrices is in echelon form (but not necessarily reduced echelon
form)?
1o 32]
a)
b)
@
d)
~ll [~ ~
[
1 0
0 2
O~
i] MATH 220 FINAL EXAMINATION
3. What geometric figure
./
::;?aV\ ~l ~
'f\ ) -'"
\l'3}
b) a line.
A L~
a plane.
-;i->.
~
\('3'
--'"
5
.~\) = [ ~
bed"'
\~
rJ As "-
N
;;L -
a) A only
we
¥\Led.
b) B only
4'~t-clJ- t3
S 0
c) A and B
@ A and C
rvr Ii;
(,v C;
[1
~
~Q
o
l1 0
3
I
0
tV Ll_
0
o~
I
':l.
db
0
<:)
\ \3
r ,\ : L
'-
0
v
\
'q
IQ
1
0
()
.;.. e.
I'n~ ~
CtLN'-WO+·
$
r
(M.-
A 5pClM1"
EY'
h ..... ·1
V ectot":r
.
tQ:3
tR3
\
J
\a)
\
\
\
~
,
0
(j)
5
0-\
'0
r.J
w], m, m, [m
SO
G-~
0
J lffi
:3 f' ;Vo t:> )
31lQ)~ ~~
,
;~ 9~ to
CQ\U ~'" G ~ A .
~I A ::: ~
L:V\.L.t.I..IrI:l-
3
u.I~ck
C
c=
A I.U~
d,..t.k;1'h ~ '-'L
d.<~ .-" ; ~0.9-
4. Which of the following sets of vectors span 1R3 ?
A=W], m, [!]}
CoL
OJJ
*,0
0\\
-\C(
~VVI
two pl'vor O-olcA.Y"'\Vl1'"' I SO
5 u It>
5"f' "'" 1iii iV; -' v:; '[ is
~
?\'\lot-
Q_\\
«)
-
A)
4
0
(A~
-rhu'-<'-
Y()..M-\~
\) ~ l \
\\
-\
V'LI-d.
?
.J
~
~ :£~
,'s
¥-2;;/3 1. \fJ e...
V'cc...tV">
./
~
I
d,VVI. u,\A \ ,. e..
d) all of 1R3.
A~ l ¥,
m'H] ,[i!]
mformed from the span of the vectors
.
a) a pomt.
(£)
PAGE 3
o
.:2..
N
CD
0
{)
0
-'1. -1-
0) ')._
'3 0Ijo-\:~/'
50 C
5pvvn~
If< '3,
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PAGE 4
FINAL EXAMINATION MATH 220
5. For what value(s) of h is the system with augmented matrix
G)
G~
=~]
consistent?
It is consistent for all real h.
is consistent only if h
c) It is consistent if h
8.
8.
d) It is never consistent.
[;
4
h
-
'3]
tr1 cd; I::Vy-
),j0
Yb
0..
)0
N
-6
Lk
h-K
0
Ulha
J
31
LI
[/
0
-t4
l;/-J-
~5~;"
;~V\L
f~M
J
[0
Go~;s~
~
h
IS- "
o
; ] .
~oV'
0
aJ\
Uf; 1/ no .J­
\fa..\lJ,..(,Y-
};;v.....
~. k
6. Suppose A is a matrix with three rows. Three of the following statements are equivalent.
Which statement is not equivalent to the others?
a) The columns of A span
jR3.
A has exactly 3 pivots.
@ A has a pivot in each column.
d) Ax = b has a solution for all b
f).-I
~
r~~
anJ c
43 .
(c,ve, Jr;n, t
/c.no vi
~/ fh~;r
&-0 I U.VVl Vl')
A(
lCvr)
E jR3.
~
~ u,;,,, cJ e·J
~
,~o
t"('
Y'I
~
0 ......
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MATH 220 FINAL EXAMINATION
PAGE 5
7. If the augmented matrix of a system of linear equations has the reduced echelon form shown
below 2 0 0 -1]
0
0000)03,
[ OOOO([) 2 then what is the set of solutions of the system? CD
?t-I)
0
a) x b)
x~
3 +xa
1
2
0
0
0
~ ~S -.
?t~>x..\.\
-2
0
1
t~
1--:=. ":.
')\..). + 2
= -\
')(.~
'Az..t -=
~S
n] m
?t\
I
:::
0
@ x=
I
0
3
2
+Xl
1
0
0
0 +xa
0
0
0
0
-1
d) x =
I
0
1
3 +xa 2
2
0
0
0
\ ~,
-\-')..)(.
-:.
:,
.
;;2..
'"iI.?
1
<>
0
-\
0
0
;t..
"X..c,
?t':i
"X.y
-1
3
=
-x.~ 'X'20
+x,
bCA..s: c
'3
;2
-t
?t\
C
0
0
Hg
\_°2~
\
PAGE 6
FINAL EXAMINATION
MATH 220
8. Find all values of h such that the set
S=W], m,
nn
,
is linearly dependent.
VI
a) 0
UJe.
(WI
fOr
c) 2
l
"2- ~"3 ~J
~
~[~
\"'\.£ (!.
1 -3
-1
(j)
o
1
\
1
1
: :-. 2
A.-I
1
01
0
0
h~J
/:;Q
d
XI
wh,'c h
Ilw!
h
:2
,I It
J
==
VI +
ha-r
~
~
?\. '2.. \( l... -t x'3 \1"'"3 ~ 0
( t:k ___
t-v"l ()
5"
+-
k
~f\-V- V~· 04 b-)
O/VL...
~] ~ [:
-J
---"
J "­
Val ues-
501Ckt-; Ob'\
-In'II;c..J
a-r
-'>
\fl
I­
'1'.3,
1("
(kin
3
~
,t
1
-1
h'i"2
3
-I
-I
-'~
?t,:1
0
0
J ~U
'0
)'>
~
1 -/
I - 'J
_/
OJ
0
h1'2
0
I ~ "~·",hl(..
.
0
9. If T : JR2 ---+ }R2 is the linear transformation which first refiects points in the xraxis and
rotates points in the counterclockwise direction through 7r /2 radians, then what is the standard matrix of T? a)
If)
b)
I ~
c)
[6 ~]
@
[~ 6]
1 I
-11
~
.....
~
\~e\
~
~
=
l ~J
el
+
-""
€..";l. ~-
~
ez'
~
=
l~j
?
/'J'
-e'2­
Sea Vld evrci
f"Y' CA-./;-
y: ~
A [7(e:;'j
<;
1i.;)J
~ \_~
~l
\
MATH 220
FINAL EXAMINATION
PAGE 7 lO. If T is the linear transformation defined by the formula T(x)
= Ax,
where A
=
2 2 '
[4123]
then which of the following statements is true? a) T is one-to-one and onto b) T is one-to--one, but it is not onto @T
is not one-to--one, but it is onto d) T is neither one-to--one nor onto [1~ ;;).)
A=
--dL1'-l)1..3
rW:l '3 1
J J L() (9 \01
31
N
i,.-- ""
~ tree
.$
lk
(JI'VC-/
"tAn
ee..<-k
YOW
rOY
bIt
11. If
A
~-:y
T
AA=O
,'s
==?
o ",--to
T
() t\..L- ;.. -to _0 ~ .
n-o-l­
/'5
[1 -1 1]
o
1-1 , 001 then what is the second row of A-I? ®
[0 1 1] b) [0 0 1] c) [-1 1 0] l'
d) [1 -1 0] ,
[A !1]
::;;:.
~ r:
1
-I\
0
o
-\
0
0
1
0
0
1 1
°1
-0
1
\()
,0
1
0
\
oQ
1
0
;j
cl
0
'0
=
I
~J
N
[~ A-I]
\ 10
'C
-I
-1
0
0
0
<0
-1
0
1
1
'V
0
1
\
PAGE 8
FINAL EXAMINATION
MATH 220
12. Which of the following is a subspace of ]R3?
a) The null space of a 3 x@matrix.
b) The column space of a {)x 3 matrix.
c) V
@
13. If A
=
W
~ { [~J
~
m:] :
3 -1
1
0
[ 7 -2
a) 0
b) 1
@2
d) 3
Xl
+x,
4Xl -
~]
~ 0}
CA 'S
--?>
---7
e 'CJ-.
IJ.J....} sf:>~c.~
su.'bS?~ ~
0...
u==[ I:]
b"""
_u =
is;.,
-I{'"
(
\\":<..11
1 -t 1 '=
hie.
Q..
l~
0 )
[=; ) '; ". !- '" -v- ( s . ~
/'
X,
~ \~l\
(-1);-<.-'1)-.0
I
= 0}
.
'Iv
c<­
-a
then what is the dimension of the column space of A?
~
OJ/II =
Y"~/c fJ
r 3 -\o ,OJ
~
A~l~ - l ).
~ L~
o
-I
-?~
2
-b
=
f'..J
# ~ pt'vo-h-
[1
::3,
0 ~1
-\
0
'1- -~
~ ~<:D
rJ
'"
-\~
p/V01S"
J A.
0
<>
~ CD
0
0
;t~6
0
­
\
MATH 220
FINAL EXAMINATION
What is the determinant of the matrix
a) 0
1
8
29
?
59
.
0
3+5h 5+2h 11+9h
'0
q
';;(
8'
()
~q 5
3t?~ 5-t'2h \\-tq~ '0
t)
120
2
o
o
5
b) 60
o
[~
5
h
t)
d) 180
o
()
10
5
@
o
()
{)
Ro.
R ':J ~J
15. If A
5
\\
®
0
'g
q
.2..cq
d
5
d­
5
0
t>
'3
~ f{e~Jt.cA.- 1<3
\\
S
©
C
C0. J
1C2> )V~ )
.::Lei
--t~
1<'-1.
J
0­
..t.Y"l·c.-~wf ~
f"l. ~'I'\.
d.-"(J e'-hd [;!6]' then what are the eigenvalues of A?
@ 3 and -7. c) 5 and 4. d) -2 and 3. =l. ~b-~
?;),
cld: ( ~ __ '.\ T)
0
(?- - >..) (- b- ).) - C\
==?-
-=-2",
=7
-':.-r
~CJ;
r';",
~r-: e~
f..-o d€A.d'
. . . ~ t:: ~ a) 2 and 3. A-AI
c~..,~~~)
('=» =- \ J 0
del;
1<. 4
ba­
R3- f<-1
(del:- do&' not­
9
2.
o
PAGE 9
-+ ,L\ A_ ~ \
0 - ";;)l A-r1)
\~
~ -= 3} - f­
C
0
Q
\
Uv.­
l;J.....R....
\
MATH 220
PAGE 10
FINAL EXAMINATION 16. Suppose A and Bare n x n matrices. Which of the following statements is always true?
a) If A and B are similar, then they have the same eigenvectors.
~1lllUd.l
If A and B are
@ If A and B
then
+ det(B).
+B)=
are similar, then they have the same eigenvalues.
d) If A and B are similar, then they are diagonalizable.
~e~
\~ e QV'E.~
L\
Q",
'3\
~c..~
S .
17. If A is the eigenvalue of an n x n matrix A and x is the corresponding eigenvector, then
which of the following statements is always true?
@ x is in the
space of A - AI.
b) x is in the null space of A.
c) x is in the column space of A - AI..
d) The column space of A - AI is all of lRn .
.
-to
.-.:.. \
e., ~ ~ \j (! c t:. 0'('
-r)-L
'\, -r-k; l'
e~
fO-'i¥L
""'­
\~
\~
e-;~ ~ S\JO-CS!602...\
~
\:::~
~
\1'"""'
e\%~~~o..ce..
Nul \
A -'\l:...-)
t~-bC>C.K
,
r~S ~ 1)
L. :~
V\.. .
\
MATH 220 FINAL EXAMINATION
PAGE 11
18. If A, Band Care 3 x 3 matrices such that det(A) = 2, det(B)
what is det (2ABT C- I ) ? 3 and det(C) = 4, then a) 3 b) 6 @ 12 d) 3/2 01 &t-( 'l.A Q,"T c \)
~ l:'
-=
a) P =
Z\.~)('3)( ~) =\';1
[ 1-1]
-2
P- \ R P
l' c) P
@P
FOr~~J.. l
A -
A- A~ ~
[-3 5]
1 2 '
D"1
51..
\"= [.1, '1'21
rl -'3A
cl.iri(4 - \ 1-) -=c)
[21 1]1 .
21.
=- [. -_':.0
oJ -: :. [- r;
_ '3
=[~
-= \.:..;,
A -= PDP-' P= 'A::::. S
e. . .-1. .j. -e\ c.
[::::~~], then which of the following is a matrix P such that p- I AP is diagonal?
19. If A
{:or
-letA dd,
ij
6.
=7
V.-~)
01
f\
bOjNl
{.
'3
b
'g-'A
(- \ -
(1-
'5)
-~
C
0
tv
A) t~ - ').) -It
='0
O}
\'g -;;:
a --=c)
A'2.- +- A -t- \'O-()
~A=- ~I 5
X=
~~ ~~
l ~j
[~1
~\ ~
l ~J
~L= l~J
112 C
0l. rL",ooJ
\ -\ 01
o
1
\
MATH 220
PAGE 12
FINAL EXAMINATION
and u = [_ ~] , then what is the distance from y to the line through u and the
20. If y =
origin?
<'
-"
/\
V5
@2V5
c) 3V5 d) 4V5
- S u...
5
rt -\]
';;;t
_:l>
=
z
=-
u..
~=
a)
I.A..
~-;,~ n~- \-~~ ~~~1 ~·51-{).Jhu.."""
11
zll
-= )
and if y =
what is
a)
Vl -
b)
2Vl-V2
@
d)
2V2
+ "2V2
1
Vl
+ "2V2
VI
=
[1]
orthogonal projection of y onto W ?
-
..>
A
~--
1
2Vl
[2;; :: 2Vs \
I b -+ '-l
21. If W is the subspace of IR' spanned by the orthogonal vectors
-===­
-'"
lJ-.1/l
V'i :Vr
...>
v.+
b -"
~ \J\
-+
\f,
~
-"
:2.
-~
8-. 'V"2­
. V.. ..L- -"
':l. \f",
.,...!!o.
\G. ~
I
.­
-v;.
and
V,
=
[-~]
\
MATH 220
22.
llxl
=
FINAL EXAMINATION
[~]
,x, =
[!] ,
md
same as the span of Xl, X2 and
.)
-@
c)
d)
1] [ 1/2] [ 1/4]
[~ , -1~2 '
-m
1/2]
-1/2
,[-1/2
1/2'
[~] TI~~] ,U]
[l [-:~:] ,U]
X3
=
[_
~]
PAGE 13
, then find an orthogonal set whose spm is the
X3'
usC':
Scl".~k
&YO-1M-
f'v\:>
ce..s.~
~
'J..\
'f\
...:..
'\['2 =
X2
-"
dt::
Y-::<.' v\ .~ =
--"
\}'2.
_\'1.
- \/"2­
\/'2­
--'"
-v;
\f~ -=- 'A-:- _
"'2- - ~\....!>.v,
'I.._?:, - V-'2 7t
""2­
\
\['2'
~
X3 -
~
0
\II
..0 \(';2.
"''4_
~
-= 'I..?,.
\
23. Find the least squares solution :it of Ax = b when A
1/3]
(]) x = [1/3
x = [~]
d)
1/3]
x = [-1/2
f1" A .,,;;
('0. '"
~ ~\(\rlA\k
_
\
'). -\
I
I,.
-=
=
m·
0
J[~11~J
1
11
=[
= [
l
1 T1
V3 J '" L
1
:t
1
A
1[1 -1]
, = J31[1v'2 -J2lJ
1[J3 -1]
c) = 2
J3
1[~2 1]
@ J5
1
1
0
r
U
A -).. ~
--""
- VUl-~'
1
ll'f'il\
=
f5
I
~J -=-- [U\ U'2-J = i5
[b -)
-'1
~ ~'2 l-; J
A-:-
x' ~
\f.. ') "'- ' [-2J
,
\1-3 j
[_~
-;]?
- '2.
'3
-1J
J
3-1
'. [A ... 11 0 J =
for A
1 2
_1
211
1
1
Q
I')
,,2.
\= QY'
NOy- Y'I~3 <.
'Is
<)
III l'
ci~-t-(f\-:\'I.)-=.c ....:..'j. (6-':\) ~->')-4:o.o
'7
-q A t- 11..\ = 0 ~(A- --;:;-) 0,.-··:).) =0
~ >--::::71::;.2
1
U=
~
PiI A
'/'3l
1/'3 J
=
1
b) U
1J
ot
0
24. What is an orthogonal matrix U that diagonalizes the symmetric matrix
v'2
2
1
rv_'1_
?\..
a) U =
1}
2-
12 111 [1 ).1 11J-vrl 1
1
,....
and b
t:v.J8 ~'" e,....,.c<d ~ yi)A
').
~<::..\-- ~ f'~"" a,l!:.Q
:]
~ (~ 1 1] rfJ
T#
I)
- "3l-\ ~ 1~ \1 l Y'3\
\! v'-\
rLO1
=
A"
;~"~\a\<
[~
A TA X-= A'lb.
A'ft
c)
=
-""
$ol~
1/2]
a) x = [1/2
(SihC"
PAGE 14
FINAL EXAMINATION
MATH 220
2... -.
?{ 'L [
"t~
_
U~"='- \\\t,\\-
[-2
_
I
1J
::2.
l
l .- '1
l..\
l- t 1
vt
8 - '2.1 0'"\
I{zJ
\" - \ - '2
.J
=-
r <-{
L-'2
.- 2
I
~ = t 1J
7s- L~]
0,,.., T \
t:>.J
0-Lv
1:)
J
'- ()
'}
0J
0 C
f 1 -1/2OJ
l 0 0 't
\
25. What is the matrix of the quadratic f9lfll5xi -
5-4]
@ [
-4 -7
b)
PAGE 15
INATION
MATH 220
[~ _~]
8XIX2 -
7x~?
for gd: ~, i
VJ ~
~ (A~
vver\--
,teS \ ' ()V\
Co'J~e~~
\
'*:--,