MATH 133 Midterm: MATH133 Fall 1998 Exam
31 Jan 2019
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Department
Course
Professor
Final Examination Deember 14, 1998 189-133A
Group 3
PART I: Multiple Choie Questions
1. For the linear system
2+a1a
4+3a3+a
!
x
1
x
2
!
=
1
1
!
:
(a) When a6=
1
2
and a6=1, there are innitely many solutions.
(b) When a6=
1
2
, there is a unique solution.
() In both the ases a=
1
2
and a=1, there are innitely many solutions.
(d) When a=
1
2
, there are innitely many solutions.
(e) When a=1, there are innitely many solutions.
2. Fora33 matrix A,we have A
3
=I, where Iis the 3 3 identity matrix. What is
(AI)
1
?
(a) A
2
A+I
(b) AImight not be invertible
()
1
2
(A
2
+A+I)
(d)
1
2
(A
2
A+I)
(e)
1
2
I
3. The point(3;2;1)
t
is on both of the planes
x+y+3z=2
and
xy+z=4:
Whih of the following gives all the points of intersetion?
(a)
x+3
2
=
y2
1
=
z1
1
, (b)
0
B
x
y
z
1
C
A
=
0
B
2
1
1
1
C
A
+t
0
B
3
2
1
1
C
A
,
()
0
B
x
y
z
1
C
A
=
0
B
6
4
2
1
C
A
+t
0
B
2
1
1
1
C
A
, (d)
0
B
x
y
z
1
C
A
=t
0
B
3
2
1
1
C
A
,
(e) Only the point(3;2;1)
t
.
0
b
1
1 0
1 2 3
1
Final Examination Deember 14, 1998 189-133A
5. Whih one is a subspae of R
4
?
(a) f(a; b; ; d)
t
2R
4
ja
2
=b
2
; +d=0g
(b) f(a; b; ; d)
t
2R
4
ja+b=0; +d=0g
() fv2R
4
j
0
B
1 1 1 1
1 2 2 1
1 0 3 1
1
C
A
v=
0
B
1
1
1
1
C
A
g
(d) f(a; b; ; d)
t
2R
4
ja
2
+b
2
>0g
(e) all vetors that are not orthogonal to (1;0;0)
t
6. What are the oordinates of the vetor (1;1;1)
t
with respet to the ordered basis
f(1;0;0)
t
;(1;1;0)
t
;(1;1;1)
t
g?
(a)
0
B
1 1 1
0 1 1
0 0 1
1
C
A
1
0
B
1
1
1
1
C
A
, (b)
0
B
1 1 1
0 1 1
0 0 1
1
C
A
0
B
1
1
1
1
C
A
,
()
2
6
4
1
1
1
3
7
5
, (d)
0
B
1 0 0
1 1 0
1 1 1
1
C
A
1
0
B
1
1
1
1
C
A
, (e)
0
B
1 0 0
1 1 0
1 1 1
1
C
A
0
B
1
1
1
1
C
A
.
7. Suppose v
1
;:::;v
6
are vetors in R
4
and we write them as the olumns of a 4 6
matrix A. Upon doing some elementary row operations, we nd a rowehelon form of
Ais
0
B
B
B
1 2 3 4 5 6
0 0 1 1 1 1
0 0 0 0 0 1
0 0 0 0 0 0
1
C
C
C
A
. Whih of the following gives a basis for spanfv
1
;:::;v
6
g?
(a) fv
1
+v
2
;v
3
;v
4
;v
5
g
(b) fv
1
;v
4
v
3
;v
6
g
() f(1;0;0;0)
t
;(4;1;0;0)
t
;(6;1;1;0)
t
g
(d) fv
1
;v
3
;v
6
g
(e) fv
1
;v
3
;v
4
;v
6
g
8. Suppose a
ij
are hosen so that det
0
B
a
11
a
12
a
13
a
21
a
22
a
23
a
31
a
32
a
33
1
C
A
=2.
What is the determinantof
0
B
a
11
+3a
21
a
12
+3a
22
a
13
+3a
23
a
21
a
22
a
23
a
31
a
32
a
33
1
C
A
1
?
Final Examination Deember 14, 1998 189-133A
9. The vetors u
1
=(1;0;1)
t
;u
2
=(1;1;1)
t
;u
3
=(0;1;1)
t
form a basis for R
3
. Applying
the Gram-Shmidt orthonormalization proess to this ordered basis yields the following
orthonormal basis of R
3
.
(a)
8
>
<
>
:
1
p
2
0
B
1
0
1
1
C
A
;
0
B
0
1
0
1
C
A
:
1
p
2
0
B
1
0
1
1
C
A
9
>
=
>
;
, (b)
8
>
<
>
:
0
B
1
0
1
1
C
A
;
0
B
0
1
0
1
C
A
;
0
B
1
0
1
1
C
A
9
>
=
>
;
,
()
8
>
<
>
:
1
p
2
0
B
1
0
1
1
C
A
;
1
p
2
0
B
0
1
1
1
C
A
;
0
B
0
1
0
1
C
A
9
>
=
>
;
, (d)
8
>
<
>
:
1
p
2
0
B
1
0
1
1
C
A
;
0
B
0
1
0
1
C
A
9
>
=
>
;
(e)
8
>
<
>
:
0
B
0
0
0
1
C
A
;
1
p
2
0
B
1
0
1
1
C
A
;
0
B
0
1
0
1
C
A
9
>
=
>
;
.
10. A oni in the xy plane is given by the equation 4x
2
2xy +4y
2
= 3. After making
the hange of variables
x
y
!
=P
X
Y
!
, where P=
11
1 1
!
, the oni has the
equation
(a) 4X
2
2XY +4Y
2
=3,
(b) 3X
2
+5Y
2
=3,
() 6X
2
+10Y
2
=3,
(d) 5X
2
+3Y
2
=3,
(e) 10X
2
+6Y
2
=3.
11. Let
A=
0
B
13 3
3 1 3
33 1
1
C
A
:
Whih of the following is true?
(a) (0;0;0)
t
is an eigenvetor for the eigenvalue 0 of A
(b) (1;1;1)
t
is an eigenvetor for eigenvalue 7 of A
() Ahas no eigenvetors
(d) (1;1;0)
t
and (1;0;1)
t
are both eigenvetors for the eigenvalue 2 of A
(e) (1;1;1)
t
is an eigenvetor for eigenvalue 1 of A
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