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Title: Introduction to Multivariable Calculus
Description: Introduction to Multivariable Calculus, calculus, exam, mocks, algebra, university
Description: Introduction to Multivariable Calculus, calculus, exam, mocks, algebra, university
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MATH 113 Introduction to Linear Algebra L4
Midterm Test
hteX IVEwrEPHHWD UXHHpm { VXQHpmF
xmeX
tudent shX
ersionX eGf
otl mrksX IHH
Answer all questions
...
1a @IS mrksA e liner system
e
x a b with the following oeient mtrixX
P
Q
I I P I
e a RP Q P H S
Q R R k
is known to e inonsistent for prtiulr b in R3 F hetermine the vlue of k F
SolutionX e perforum row redution on the oeient mtrixX
P
Q
P
I I P I −2r1 +r2 I I
R P Q P H S
3 R H I
−3r1 +r3
H I
Q R R k
P
P
P
I
P
Q
P
I I
S R H I
3
k Q
H H
−r2 +r3
P
P
H
I
P
k
Q
I
S
sf k Ta ID the oeient mtrix will hve pivot position in every rowF gonsequentlyD
e onsistent for every b in R3 F
e
x a b will
ine it is given tht for some b in R3 the system will e inonsistentD so we must hve
Qn
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us
t
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ust
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2a @PS mrksAX he following is n ugmented mtrix of liner systemX
P
I
T P
T
R I
Q
I Q
Q U
I Q
R IH
Q
P
S
I
V
I
PU
UX
IS
S
@A rnsform the mtrix into its redued row ehelon form @ipAF
@A olve the orresponding liner system nd express the solution in prmetri vetor formF
SolutionX
@A fy row redution lgorithmD
P
I
T P
T
R I
Q
I Q
Q U
I Q
R IH
P
S
I
V
Q
P
I
I
P U −2r1 +r2 T H
U
3 T
I S r1 +r3 R H
S −3r1 +r4 H
I
I
H
I
Q
I
H
I
Q
P
I
I
H U −r2 +r4 T H
U T
3 R H
PS
H
P
P
I
I
P
I
I
H
H
Q
I
H
H
Q
I
HU
U
PS
P
P
I
I
I
with further row redution to ip @kwrd phseAD
P
I
TH
T
RH
H
I
I
H
H
Q
I
H
H
P
I
I
H
P
Q
I
I
U −2r3 +r1 T H
HU
3 T
P S −r3 +r2 R H
H
H
I
I
H
H
Q
I
H
H
P
Q Q
I
U −r +r T H
P U 3 T
H
H
I
H
2
P
H
1
S
H
I
RH H
H H
P
I
H
H
H
H
I
H
I Q
P U
U
P
H
S
@in ipA
X
@A he liner system will hve the sme solution set s the following liner systemX
V
b x1
b
`
x2
C Px3
C x3
b
b
X
et
x3
a I
a P
x4 a P
HaH
a sD then the ove system hs the following the generl solutionX
V
b x1
b
`
x2
b x3
b
X
x4
a I Ps
a P s
as
aP
where
Y
s
P RF
reneD the solution of the liner system in prmetri vetor form isX
I Q
U
T P U
2U
aT UC
P
R
I
H
S
Y
where
s
P RF
gx
x/
s
cs
_
S
k/
~
x4
H
P
om
e
...
h
R
3
ih
Tx
T
Rx S
P Q
T I U
T
U
P
://
Q
x1
ht
tp
P
P
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3a @PS mrksA
@A se row opertion methodD nd the inverse of the following mtrixF
P
Q
I I H
e a RI H ISX
H I I
@A e mtrix trnsformtion
P
X R3 3 R2 is known to stisfyX
Q
I
RIS a
H
pind the stndrd mtrix
P
I
f
P
Q
!
I
I
RHS a
Y
Q
I
!
Y
P
Q
H
nd R I S a
I
P
!
X
Q
of F
xoteX he ove onditions n e omined s mtrix equtionF
SolutionX
@A e perform row opertions on
P
I I H
RI H I
H I I
P
I
3 H
1
r
2 3
H
P
I
−r2 +r1
3 R H
H
−r2
R
Q
I
H
H
I
I
H
H
I
H
e
j
s3
X
P
H H −r1 +r2 I I H I
I H S R H I I I
3
H I
H I I H
Q
P
H
I H H r3 +r2 I
I I I H S 3 R H
1
I 2 1 1
H
2
2
Q
1
1
1
H 2
2
2
1
1
H 1 2 2 S
2
1
1
1
I 2 2
2
P
rene we hve
e
−1
a
1
2
R 1
2
1
2
1
2
1
2
H
I
H
I
I
H
Q
P
Q
H r2 +r3 I I H
H S R H I I
3
I
H H P
Q
H I H H
H 1 1 1 S
2
2 2
I 1 1 1
2
2
2
I H H
I I H S
I I I
1 Q
2
1 S
F
2
1
2
1
2
@A he given ondition n e restted sX
P
Q
I
f RIS a
H
P
I
P
!
Y
Q
!
I
I
f RHS a
Y
Q
I
P
Q
H
f RIS a
I
P
!
X
Q
sing the denition of mtrix multiplitionD we n rewrite the ove sX
P
Q
I I H
f ¡ RI H IS a
H I I
P I
I Q
P
Q
!
X
1
2
1
2
1
2
1 Q
2
1 S
2
1
2
a
1
2 3
2
1
1
2 2 7
2
5
2
R
!
cs
_
1
2
k/
~
1
2
R 1
2
om
e
...
h
P ¡
Q
P
ih
P I
a
I Q
!
://
Q−1
ht
tp
f
P
I I H
P I P R
S
a
I Q Q ¡ I H I
H I I
!
gx
x/
reneX
X
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4a @PH mrksA
@A pind suitle invertile mtrix
suh tht
e
is in ipF
P
Q
I P I I
Q I PSX
e a R P
I I P I
@A ixpress the generl solution of the following liner system in terms of
V
`
x1 C P x2 C
x3 C
x4 a
Px 1 C Q x 2 C x 3 C P x 4 a
X
x 1 C x 2 C P x 3 C x 4 a
X
SolutionX
@A erform row opertions on
e
P
j
s3
X
Q
I P I I I H
R P
Q I P H I
I I P I H H
P
I P I
3r2 +r3
R H I I
3
H H H
P
I P I H
−r3 +r1
RH I I H
3
H H H I
P
Q
H −2r1 +r2 I P I I I H H
H S R H I I H P I H S
3
I r1 +r3 H Q Q P I H I
P
Q
Q
1
I I H H
I P I I I H H
r3
2
H P I H S R H I I H P I H S
−r 3
2
P S Q I
H H H I 5 3 1
2
2
2
Q
P
7
1
3
2 1 −2r2 +r1 I H I H 2 1 1 Q
2
2
2
2
P I H S R H I I H P I H S
3
5
1
5 3 1
H H H I 2 3
2
2
2
2
2
hereforeX
P
ke
1
2
aR P
5
2
1
2
I
3
2
1 Q
2
H
P
I H
RH I
then e a
H H
SY
1
2
I
I
H
Q
H
H S is in ipF
I
@A he oeient mtrix of the system is the sme s the mtrix e in @AF ine
we hve ex a b 6 ex a bD nd the system ex a b isX
P
I H
RH I
H H
Q
I
H
P
Q
a sD then the generl solution of the system in prmetri form isX
Cs
s
s
s
PR
X
gx
x/
1
2
where
cs
_
x4
k/
~
b x3
b
X
1
2
om
e
...
h
x2
a
a
a
a
ih
V
b x1
b
`
://
x3
is invertileD
1
H
2
HSx a R SX
I
1
2
ht
tp
et
I
T
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5a @IS mrksA
@A ivlute the following determinntX
Q
P
H
I
I
I
P
S
H
H
H
P
Q
S
X
R
P
@A olve the following homogeneous systemX
V
b
b
`
b
b
X
Qx 1 C
Px 1
x1
x2
x2
Px 2
C S x2 C Px3
C
C
C
C
Q x4
S x4
R x4
P x4
a
a
a
a
H
H
H
H
SolutionX
@A fy oftor expnsionD we hveX
Q
P
H
I
I
I
P
S
H
H
H
P
Q
Q
S
a P ¡ @ IA4+3 P
R
H
P
I Q
I S
P R
I
I Q
S
C P ¡ @ IA2+1
g
a @ PA ¡ fQ ¡ @ IA
P R
P R
a @ PA ¡ fQ ¡ @ R IHA P ¡ @R TAg a UTX
1+1
@A he oeient mtrix of the system is extly the sme s the mtrix in @AF ine it hs
nonEzero determinntD the system will hve unique zero solutionD iFeFX
V
bx
b 1
`
x2
b x3
b
X
H
H
H
H
ht
tp
://
ih
om
e
...
h
k/
~
cs
_
gx
x/
x4
a
a
a
a
V
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Title: Introduction to Multivariable Calculus
Description: Introduction to Multivariable Calculus, calculus, exam, mocks, algebra, university
Description: Introduction to Multivariable Calculus, calculus, exam, mocks, algebra, university