of M
ETU
Department Mathematics
MATH 260
SECOND MIDTERM EXAM
Date
Time
:22.11.2010
Duration
: 10A min.
:17.10
Name
Student ID
Section
Signature
3 Quetions on 4 Pages
100 Points
SHOW YOUR WORK
Q.1. (30 pts) a) (5 pts.) Show that the foliowing functions
x,I*8, r* sin"r, fro - t, and r*coszs defined on IR are linea.rly dependent.
x + ( t tx) -(x a sfr'*) - fi,n bsux)
b) (10
V:
-;
i:
j',
I
;I
=a,
(g,t.r6*)=11'=o
pts.)
Compute the dimension of the vector subspace
span{( -L,2,9,0), (5,4,9,0), (3, 1,0,0)} of lRa.
t;- G,,", ?r""l'-3. y g ffi 4
; bJ fferLu + 3 oi wu"L'
c) (15
pts.)
/",*(V) =z
oo o)
Find a basis for the row space of ,4 and find the dimension of the row
space of .4.
tooav
lo t -,t o lv
l6o zo t f'r
lszs: t J"
[-To
1\-
lll;llffiEil\\lH'
et2 t,\ {(, ot or) , b , -t t o) ,(ooz sr), (ooo' '\
l3*
d,t*q-h9ruh
),
-
q.2.
(35
pts.) a) (SaS-10 pts.)
Evaluate the following determinants (no
pa.rtial credits will be given for an incorrect numericai answer to the question Q.2.a)):
12042
23456
02021
00030
L2041
r2345
02040
54321
00012
0003
b) (15 pts.)
11
z
=ofr;WIi=(q)'li,i I {rl'l ,':l-'l
:ep)(r. (t) -r' o) = 2
(
L{
=
4
li i?\
t;?\-'ll;
Compute the adjoint (adjugaie)
r
o: |l-111.1
l2 ? 3
1
r-sLz{l
I
of
'll;
I
,td'() = \3
l=
!
I'lr-['l zO[=-J a ,;air'li 3
?,,=("')'r Iti'ul =-S at2\'/ rqal la''
u''=
tr
[lll=
-s
t,f.'li o1z ez,= li, Ll=-7 a,{- l;ll = 7
a'f-l;i\=o Qz3=lill=2
c't-- Uli=t
er,=
-
A,=fii:il ogl+=*E-;]1
g.u^)-
,4=
*'a?* ,rrn) --t^l:;'",tlot =
zl','ol
,, (A)--ttt -o,\=
h3'-i
-tl;il=
WqA1;ilfr??li;:1i,1'*Elih:';1'13'
-ti:!J*??)ff
;Bi'Ai+fl*l;il't?il:r,'{!,1A'
d A'=(;) rl'fl:i]= EiA
c) (saf-10 pts.)
Given that
\a1
blb
a2b
lL+a
a-fl a*2 b+f
brb
=21 s
le
2a42b
12 ar3
bx s2
ar 2r2
:
8tZ
7
a
z
q(
x4
brg
brs
= x'x2-
,t l;
lq
I
2
9, compute the following determinants:
4l
t*81 lr q el=2.3=
i''l=zle L
l8
4 / l" z dl
{i
=
6 x lral' e\= 9xt
Gl=yu,
4
sl
\
la
la Q s
I
Q.3.
(35
pts.) a) (fO pts.)
Exbend {L
t
fro o o
lo r
l,0oo
+ r2,r
Bas,g
-
r,3}
to a basis for the
'b
{
{,,,',,-r',
o
lo-r \-/ Jo
N\-l \J
J
i\
*
d
Lt
?
Lru)
X
vv
b) (L0 pts.)
Find coordinates (the coordin-ate matrix [u]6') of LL: r - s2 + as with
respect to the basis C : {wt,w2,u3]1 of the vector space il/ : span{uhtl!r2trl3}, where
il)r : f, * t2, W2 : fr - X2, and Uh : r + X2 + 2r3.
t= wrt x3 wr'r
: - !*, J w'* *
=
!(w;w,)
.+
=X-X+x
1-t
I -,
--
/rn(= | .L-Jr-I1
L,
c) (5 pts.) Let to e W be such that lr1,
defined in
v,/
b). Find tbe polynomial u.
:
Ii ]
I
I
)
where
c
is the basis for
Compute the transition matrix P : PB-c from the basis B
for W to the basis C : {tlr, wz,wt} for W defined in b) .
* w.
)
lo1
,=
it.l/, -4a1'
Lrrl
ft"t" cz
9*d
su&L,ou
t o o f*fR=
ro
2
y7=
w,
I
I ,-l
Lo0
\-/ v
W, wz
I
(
L
]I/
: {r,x',rt}
-4 + bJ; F*1"til;il, r[r1r=
?: L0 -o I;rl
ltii ri'
-t
*,
+ 2(x -v') =3x - v2
= l'w rt zwzl o vr= k+x2)
d) (10 pts.)
2X = w,
space of
;_ L-t
h]
I
o
00
t/1 ;
;,-v"
\rZ
,"?:Jryqiz' 'tli'!i
wilil^fl;,i
'33i\-'E,w