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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 9/20; Due 9/28
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: Norms.
Show that for
x
∈
R
n
,

x

∞
≤ 
x

2
≤
√
n

x

∞
.
Problem 3: Continuity and Linearity.
Show that any linear map between Fnite dimensional vector spaces
is continuous.
Problem 4.
Prove that the induced matrix norm
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 Fall '10
 ClaireTomlin
 Electrical Engineering

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