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EE221A Section 10
11/04/11
1
Direct sum of subspaces
Exercise 1.
Show that if
V
=
V
1
⊕
V
2
⊕ ··· ⊕
V
n
, then
V
i
∩
V
j
=
{
θ
}
for
i
6
=
j
.
Exercise 2.
Let
M
and
N
be two subspaces of
V
. Let
{
m
1
,...,m
p
}
be a basis
for
M
, and
{
n
1
,...,n
k
}
be a basis for
N
. Show that
V
=
M
⊕
N
if and only if
{
m
1
,...,m
p
,n
1
,...,n
k
}
is a basis of
V
.
Exercise 3.
Show that if
V
=
M
⊕
N
, we can construct a linear operator
P
(called
the projection on
M
parallel to
N
) satisfying
P
2
=
P
(idempotent),
R
(
P
) =
M
, and
N
(
P
) =
N
.
2
Jordan form
Recall:
Minimal polynomial
ˆ
ψ
A
(
s
)
of
A
is the polynomial of least degree such that
ˆ
ψ
A
(
A
) =
θ
n
×
n
.
The minimal polynomial divides the characteristic polynomial
: This means
that
ˆ
ψ
A
(
s
) = (
s

λ
1
)
m
1
(
s

λ
2
)
m
2
···
(
s

λ
σ
)
m
σ
, where
σ
is the number of distinct
eigenvalues of
A
, and
m
i
≤
d
i
,
i
= 1
,...,σ
, where
d
i
is the degree of the
i
th distinct
eigenvalue in the characteristic polynomial.
Theorem:
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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

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