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EE221A Linear System Theory
Problem Set 3
Professor C. Tomlin
Department of Electrical Engineering and Computer Sciences, UC Berkeley
Fall 2010
Issued 9/22; Due 9/30
Problem 1.
Let
A
:
R
3
→
R
3
be a linear map. Consider two bases for
R
3
:
E
=
{
e
1
,e
2
,e
3
}
of standard basis
elements for
R
3
, and
B
=
1
0
2
,
2
0
1
,
0
5
1
Now suppose that:
A
(
e
1
) =
2

1
0
,
A
(
e
2
) =
0
0
0
,
A
(
e
3
) =
0
4
2
Write down the matrix representation of
A
with respect to (a)
E
and (b)
B
.
Problem 2: Representation of a Linear Map.
Let
A
be a linear map of the ndimensional linear space
(
V,F
) onto itself. Assume that for some
λ
∈
F
and basis (
v
i
)
n
i
=1
we have
A
v
k
=
λv
k
+
v
k
+1
k
= 1
,... ,n

1
and
A
v
n
=
λv
n
Obtain a representation of
A
with respect to this basis.
Problem 3: Norms.
Show that for
x
∈
R
n
,
1
√
n

x

1
≤ 
x

2
≤ 
x

1
.
Problem 4.
Prove that the induced matrix norm:
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This note was uploaded on 12/09/2011 for the course EE 221A taught by Professor Clairetomlin during the Fall '10 term at University of California, Berkeley.
 Fall '10
 ClaireTomlin
 Electrical Engineering

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