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can I also please have help on #25- 30 please please? Linear Algebra
Math 636
Midterm Quiz
1.
The geometric interpretation of the following set
W
=
±²
x
1
x
2
³
∈
R
2
|
x
1
+
x
2
= 2
´
is
(a) a point in
R
2
.
(b) a line in
R
2
passing through the origin.
(c) a line in
R
2
not passing through the origin.
(d) all of
R
2
.
2.
The geometric interpretation of the set
S
of all vectors
~x
∈
R
4
that are orthogonal to
1
0
1
-
1
,
0
2
1
1
, and
1
1
1
-
2
is
(a) a point in
R
4
.
(b) a line in
R
4
.
(c) a plane in
R
4
.
(d) a hyperplane in
R
4
.
3.
Let
~v
1
,~v
2
,~v
3
,~v
4
be non-zero vectors in
R
4
such that the solution set of the system
µ
~v
1
~v
2
~v
3
~v
4
~
0
¶
is a hyperplane. What is the geometric interpretation of
Span
{
~v
1
,~v
2
,~v
3
,~v
4
}
?
(a) A point in
R
4
.
(b) A line in
R
4
.
(c) A plane in
R
4
.
(d) A hyperplane in
R
4
.
4.
Which of the following is a scalar equation for the plane in
R
3
with vector equation
~x
=
s
1
-
1
0
+
t
1
2
3
+
1
1
1
,
s,t
∈
R
(a)
-
x
1
-
x
2
+
x
3
=
-
1
(b)
x
1
+
x
2
+
x
3
= 1
(c) 3
x
1
+ 3
x
2
+ 3
x
3
=
-
1
(d)
x
1
-
x
2
= 6
1
2
5.
Which of the following is a vector equation for the plane in
R
3
with scalar equation
x
1
-
x
2
+ 2
x
3
= 0.
(a)
~x
=
s
1
0
2
+
t
0
-
1
2
,
s,t
∈
R
.
(b)
~x
=
-
1
1
1
+
s
2
0
-
1
+
t
2
2
0
,
s,t
∈
R
(c)
~x
=
s
1
-
1
2
+
t
2
-
2
4
,
s,t
∈
R
.
(d)
~x
=
1
1
-
1
+
s
1
1
0
+
t
-
2
0
1
,
s,t
∈
R
6.
Let
A
=
1
8
7
-
7
1
-
1 1
1
1
-
1 3
1
-
3
-
3 4
1
. Then
(a) det
A
=
-
24.
(b) det
A
= 84.
(c) det
A
= 24.
(d) det
A
=
-
84.
7.
If
T
=
x
1
x
2
x
3
∈
R
3
|
x
1
+ 2
x
2
= 3
x
3
, then
(a) dim
T
= 1.
(b) dim
T
= 2.
(c) dim
T
= 3.
(d)
T
is not a subspace of
R
3
.
8.
If
U
= Span
{
3 +
x
+ 3
x
2
,
1 + 2
x
+
x
2
,
1
-
3
x
+
x
2
,x
}
is a subspace of
P
2
(
R
), then
(a) dim
U
= 1.
(b) dim
U
= 2.
(c) dim
U
= 3.
(d) dim
U
= 4.
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