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ECE 130C HW5 Solution
May 26, 2008
1. (a) The incidence matrix is:
A
=

1
1
0
0

1
0
1
0
0

1
1
0
0

1
0
1

1
0
0
1
0
0

1 1
Entry (
i,j
) in the incidence matrix relates edge
i
with node
j
of the graph.
(
1 0 0 1

1 0
)
,
(
0 0 1

1 0 1
)
,
(
0 1 0 0

1 1
)
are the
vectors that span
N
(
A
T
).These come from sending the currents around each of
the loops.
(b)
C
(
A
) :
r
= 3, the ﬁrst 3 columns are a basis.
N
(
A
) : (1
,
1
,
1
,
1) is a basis.
C
(
A
T
) :
r
= 3, rows 1,2,4 are a basis.
N
(
A
T
) : the three vectors in part(a) are
a basis.
2.
z
= (
x
+
y
)
/
2
⇒
Az
= (
Ax
+
Ay
)
/
2
3. (2 + 3
t
)1 = (2 + 3
t
), (2 + 3
t
)
t
= 2
t
+ 3
t
2
,(2 + 3
t
)
t
2
= 2
t
2
+ 3
t
3
, (2 + 3
t
)
t
3
= 2
t
3
+ 3
t
4
T
=
2 0 0 0
3 2 0 0
0 3 2 0
0 0 3 2
0 0 0 3
4. Let
p
(
x
)
,q
(
x
)
±
S
⇒
Z
1
0
(
cp
(
x
) +
dq
(
x
))
dx
=
c
Z
1
0
p
(
x
)
dx
+
d
Z
1
0
q
(
x
)
dx
= 0.
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 Summer '10
 VolkanRodoplu

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