LINEAR ALGEBRA EXAM 3 version 1 name:
1. Use eigenmagic to solve the following system of differential
equations:
dx
dt
= −4x− y − 2 z
dy
dt
= 5 y
dz
dt
= 6x+ 3 z
o
o
A y(t) = c1e(8 t) + c2et + 72 c3
B x(t) = c3
C x(t) = c1e(5 t) + c3e(−t) + c2
D y(t) = −15 c1e(5 t)
E z(t) = c1e(8 t) − 12 c3
F z(t) = 3 c1e(5 t) − 32 c3e
(−t) − 2 c2
G None of These
2. Given A =
XXXXXXXXXX
7 0 −4
, find a matrix P such that P−1AP
is a diagonal matrix
o
o
A P =
1 1 1−1 −25 −1
−8 −85 −1
B P =
1 1 0− XXXXXXXXXX
1
5 −1 0
C P =
1 1 11 −2827 1
1
2
7
9 −
7
2
D P =
0 1 11 −2 −207
0 1 2
E None Of These
3. Suppose linear transformation T : R2 → R2 with T
[
−5
1
]
=
[
1
0
]
and T
[
2
−1
]
=
[
−1
3
]
Select the true statement/s
o
o
A
T
[
1
0
]
=
[
−13
2
]
T
[
0
−1
]
=
[
14
3
−25
]
B
T
[
1
0
]
=
[
0
−1
]
T
[
0
−1
]
=
[
−1
5
]
C None of These
4. Suppose V =
{[
a
a
]
: a ∈ R
}
with the standard addition and
the standard scalar multiplication by real numbers. Which Ax-
ioms are satisfied?
o
o
A
IDENTITY LAW for SCALARS
holds
B
CLOSURE LAW for SCALARS
holds
C
ASSOCIATIVE LAW for SCALARS
holds
D V is a
COMMUTATIVE GROUP
E
DISTRIBUTIVE LAW
for scalars holds (scalars distribute)
F ALL 10 axioms hold, V is a vectors space over R
G
DISTRIBUTIVE LAW
for vectors holds (vectors distribute)
H None of These
5. Consider the following system of equations. Then determine the true statement/s
XXXXXXXXXX
2 −5 1 −5
·
x
y
z
w
=
44
−3
o
o
©2019 daabz.com LINEAR ALGEBRA EXAM 3 version 1 page 1 of 3
http:
www.daabz.com/editQ.html?nID=2jw
http:
www.daabz.com/editQn.html?coursenid=1N&v=1&TaskType=exam&QuesNo=1&key=3
http:
www.daabz.com/editQ.html?nID=2jI
http:
www.daabz.com/editQn.html?coursenid=1N&v=1&TaskType=exam&QuesNo=2&key=3
http:
www.daabz.com/editQ.html?nID=2j6
http:
www.daabz.com/editQn.html?coursenid=1N&v=1&TaskType=exam&QuesNo=3&key=3
http:
www.daabz.com/editQ.html?nID=2hQ
http:
www.daabz.com/editQn.html?coursenid=1N&v=1&TaskType=exam&QuesNo=4&key=3
http:
www.daabz.com/editQ.html?nID=2i
http:
www.daabz.com/editQn.html?coursenid=1N&v=1&TaskType=exam&QuesNo=5&key=3
LINEAR ALGEBRA EXAM 3 version 1 (page 2/ 3)
A The Homogenous solutions are given by x⃗h = α
1
0
0
1
+ β
0
1
1
1
B ALL solutions are of the form x⃗ = x⃗p + x⃗h
C The Homogenous solutions are given by x⃗h = α
1
1
−2
−1
+ β
0
5
−5
−6
D A particular solutions is given by x⃗p =
3
0
1
2
E ALL solutions are of the form x⃗ =
3
0
1
2
+ α
1
1
−2
−1
+ β
0
5
−5
−6
F None of These
6. Determine the true statement/s about matrixM where
M =
1 − XXXXXXXXXX −1 −1
−1 0 1 1
o
o
A A basis for the row space ofM is given by the non-zero rows in
1 0 0 − XXXXXXXXXX
0 0 4 −17
B A basis for the column space ofM is given by the non-zero columns in
XXXXXXXXXX
0 −1 0 0
C A basis for the column space ofM is given by the non-zero columns in
XXXXXXXXXX
0 0 1 0
D rank ofM is 4
E rank ofM is 2
F rank ofM is 3
G A basis for the row space ofM is given by the non-zero rows in
1 0 −1 −10 6 −4 −3
0 0 0 0
H rank ofM is 1
I None of These
7. SupposeB =
{(
−2
−2
)
,
(
4
4
)}
and V be the space spanned
y linear combinations of B using real number scalars. In
other words
V =
{
α
(
−2
−2
)
+ β
(
4
4
)
: α, β ∈ R
}
with the standard addition and the standard scalar multipli-
cation. Select true statement/s
o
o
A V is a vector space over R
B
(
−18
−18
)
∈ V
C
(
0
−1
)
∈ V
D
(
1
−3
)
∈ V
E
(
1
3
)
∈ V
F None of These
8. Suppose V is defined to be the space spanned by the column of the matrix A =
0 1 21 −3 −1
3 1 −1
. Find the orthonormal basis
for the space, and construct a matrix out of such basis as columns.
o
o
A
0 111
√
11 5
√
2
55
1
10
√
10 − 311
√
11 32
√
2
55
3
10
√
10 111
√
11 −12
√
2
55
B
XXXXXXXXXX −3 2755
3 1 − 955
©2019 daabz.com LINEAR ALGEBRA EXAM 3 version 1 page 2 of 3
http:
www.daabz.com/editQ.html?nID=2im
http:
www.daabz.com/editQn.html?coursenid=1N&v=1&TaskType=exam&QuesNo=6&key=3
http:
www.daabz.com/editQ.html?nID=2hY
http:
www.daabz.com/editQn.html?coursenid=1N&v=1&TaskType=exam&QuesNo=7&key=3
http:
www.daabz.com/editQ.html?nID=2iK
http:
www.daabz.com/editQn.html?coursenid=1N&v=1&TaskType=exam&QuesNo=8&key=3
LINEAR ALGEBRA EXAM 3 version 1 (page 3/ 3)
C
XXXXXXXXXX −3 4255
3 1 − 1455
D
85
167
√
167
55
1
11
√
11 5
√
2
55
42
167
√
167
55 −
3
11
√
11 32
√
2
55
− 14167
√
167
55
1
11
√
11 −12
√
2
55
E none of these
9.
v⃗ · v⃗ = ||v⃗||2
o
o
A false B true
©2019 daabz.com LINEAR ALGEBRA EXAM 3 version 1 page 3 of 3
http:
www.daabz.com/editQ.html?nID=GH
http:
www.daabz.com/editQn.html?coursenid=1N&v=1&TaskType=exam&QuesNo=9&key=3