F15 Exam 1 Problem 1
[Reserved for Score]
Linear Algebra, Dave Bayer
testlalpl
Te s t l
Name,
U n i .
[1] Solve the following system of equations.
V"
0
H
-r
—
r1
11 2
N
(
O
C
3
1
w
5
_1 2 3 5 8_
—a>\
4
X
H
6
V
z
sum of (Dre\J\OUS +^^0
w
V^v\\C 'Z_
*Z
0
u
y
,0 6-i^
0
z_
(r
'\ \ 1
oA \
0
\
=
X
howvog
C76
1 \o
'l
V '
S-2-3
' 1"
V '
H
r
0
w
0
X
1
+
1
.
0
0
1
0-1
.0.
z _
O
1
-1
1
V
0
1
0
-1_
Check:
1■
0
11 2
11 2
1
3
2
3
5
3
0
5
0
8
1
0.
—
1I
1
-1
+
0
_
0
o
1
\
o
0
r
1
1
s
-1
1
0
-1_
' 2 '
=
4
6
_t_
y
+
0
o
o
r
2
0
0
0
s
4
0
0
0
_t_
6
F15 Exam 1 Problem 2
[Reserved for Score]
Linear Algebra, Dave Bayer
testlalpZ
Test 1
[2] Using matrix multiplication, count the number of paths of length ten from x to z.
"2^
- fi W T i
^
number of paths =
"1
8 8
paths (we may ignore w)
"2-
'
A '
A
|v\
0
0"
1
0
1
0
1
1_
A^
=
0"
1
0
5 5
3 4
55
8 8
5 5
89 _
^100 n
< oc
\ 0 \
■2. a y \
H
11
\
\ zJ
ovjt
V 0 0
^^- 2 .^^
^ t- ^
^-5, 5 J t-,-I-irao 3
<y)\^ ne€^-f\r?.-)-C5luroir^
CH,3,5)'0.^,'iy 4+9^2/^-35
p\
\0
F15 Exam 1 Problem 3
[Reserved for Score]
Linear Algebra, Dave Bayer
testlalpS
Test 1
[3] Express A as a product of four elementary matrices, where
A
"2
'
1"
H
r
o
3
3
1
5
1
=
0
2
1
5
3
1
0"
-1
M
. 1-1 o. L\ o
r'°l
r \ o
\.o-i
"1
0
1
2"
o
—
T(
1
1
0
O l j
f\-z\
o \ J
F15 Exam 1 Problem 4
[Reserved for Score]
Linear Algebra, Dave Bayer
testlalp4
Test 1
[4] Find all 2 x 2 matrices A such that
A
1
2
2
3
\o"]
f^\(o)
A
=
'2
0"
3
0
"2
+
- 1 "
0
0
s
P2lf \ 1
+
+
O
o
o
2
-1
"folfdV
s
+
ioi
.r2y( IJ_
Check:
"2
3
[
0"
0
I
+
a b
c 6
2
o
-1'
o
s
+
o
o
\
1
2
2
-1
V
2
3
[Z]
"0"
0
+
0
s
+
' 2 '
t
0
=
3
a +2VD =1.
f t }
b
J
§
—
'
0
9
0J
^-2
I
0
LOJ
0^
0
- z
\
t
F15 Exam 1 Problem 5
[Reserved for Score]
Linear Algebra, Dave Bayer
testlalpS
Te s t 1
[5] Find the intersection of the following two affine subspaces of
w
X
+
1
0
0
-1
1
0
y
0
z
0
y v
0
X
1
2
y
-1
§?-§
1
0
0
-1
- 1
2
+
u
1
3
z
z,00
0
" w"
X
y
z
r z i
-3
o
,3_
2 "
w
X
- 3
y
0
3_
z
[ m m
fu)
H
7 .
riu\ = Crt
\
1 0
Q + ZU. -Z =?> u=-2.
0
1
\
- z .
u.
02
2.