Math 2270 Practice Exam 1
-
University of Utah
Fall 2012
Name:
i<ey
1. (25 points) Vector Basics
For the vectors
41
a=( 2)
3)
b=f
6)
answer the following:
(a) (3 points) a + b
=
1
/2
c=f 1
a=f 2
3)
(b) (3 points) 2a
=
(c) (5 points) cH
I
b=f 4)
\6)
=
=
•HEI7
2
/2
1
a=( 2)
3)
b=r( 4)
\6J
/2
c=f 1
(d) (5 points) a b =
(e) (5 points) Give the components of a unit vector in the same di
rection as b.
lI1l
a=(2)
\3J
c=( i)
\2j
6
b=4
/2)
?
3
(f) (4 points) Do the three vectors span a line, a plane, or all of 1R
Zc
end.
Hiec/y
C
c5e
No
a
C
/
2. (15 points) Matrix Basics
For the matrices
‘
2 4)
/i 34
B=(231)
\1 2 3
answer the following
(a) (3points)A+B=
KI)
4
—
Md
*o
L
i 3 4\
B=( 2 3 i)
\1 2 3
1
A=(
24)
(b) (3 points) A + C
=
Li]
I L(
(c) (3 points) AC
5
n
7i
)
OYO/
=
iY-
5
xl?
2 iN
3 0)
2 1 3\
)
124
A=(
(d) (3 points) AB
(
(1
231)
B=
123
=
/ t
I
(e) (3 points) BA
5
)
=
I)
6
-
4
I)
/
)
3. (20 points) Systems of Equations
Use elementary row operations to convert the system of equations
—
2y
+ 3z
—x + 3y
2x
—
5y
=
=
+
5z
=
9
—4
17
into upper-triangular form, and then use back-substitution to solve
for the variables x, y, z. Be sure to show all your work.
Y-Zy
/
-y
—,
7
5
X-y+3
y4
-
CI
4
y
7(-().
D
7
(L)3
LI
u
-
Cl)
-
—4
0
-4
0
-4
0
0
J)
—4
Cl)
>Cl)
4-•
II
+
II
II
LCD
+
LCD
I+1
I.
U
0
U
>c
>
——
oç
—
—
i
—
N
c
‘Q
‘D
I
J
Jr
r
H7 1
OrJ
—
sQ
—
0
—
rJ
-
H
5. (20 points) LU Decomposition
Find the LU decomposition of the coefficient matrix for the system
of linear equations
2y + 3z
x
—x + 3y
2x
5y + 5z
=
—
—
3)
=
=
9
—4
17
ro1
/ ic o/ (oo /(o
01 )Oo!)
/
(a
Otoo
2i
—
Jcz!
/ I
(0
10
00 (
—
H
)
I
(t1C)
00
1Lo i
L
(00
bc
(J J()
/too
1 Q
J:
2-1 (i(oo)
/(55J