Math 104 : Midterm
Instructions:
Complete the following 4 problems. Remember to show all your
work. No notes or calculators are allowed. Please sign below to indicate you
accept the honor code.
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Problem
1
2
3
4
Total
Score
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View Full DocumentProblem #1.
(20 pts) Let
w
1
,
w
2
and
w
3
be three vectors in
C
3
. Let
v
1
=
w
1

w
3
,
v
2
=
w
1
+
w
2
,
v
3
=
w
1
+
λ
w
3
,
and
v
4
= 2
w
1
+
w
2

w
3
.
Where here
λ
∈
C
. For what value
λ
0
is it always true that when
λ
=
λ
0
,
v
1
,
v
2
,
v
3
and
v
4
never span
C
3
. Justify your answer. (Hint: Rewrite the
problem using matrices).
Answer:
Let us set
V
=
±
v
1

v
2

v
3

v
4
²
and
W
=
±
w
1

w
2

w
3
²
Then we have
V
=
WA
where
A
=
1
1
1
2
0
1
0
1

1
0
λ

1
The
v
i
do not span
C
3
when and only when
dimR
(
V
)
≤
2. By the rank
nullity theorem this occurs when and only when
dimN
(
V
)
≥
3. Notice that
N
(
A
)
⊂
N
(
V
) and so
dimN
(
A
)
≥
dimN
(
V
). Applying the Gaussian elim
ination algorithm to
A
one arrives after a sequence of row operations to
A
0
with
A
0
=
1
0
1
1
0
1
0
1
0
0
λ
+ 1
0
Notice if
λ
+ 1 = 0 then
rref
(
A
) has 2 pivots. Otherwise
rref
(
A
) has 3 pivots.
In the former case,
dimN
(
A
) =
dimN
(
rref
(
A
)) = 4

2 = 2 while in the latter
dimN
(
A
) =
dimN
(
rref
(
A
)) = 4

3 = 1. In particular,
λ
0
=

1 always ensures
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 '09
 Math, Linear Algebra, Sin, Cos, WI, sin θ cos

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