MAT067
University of California, Davis
Winter 2007
Homework Set 9: Exercises on Orthogonality and Diagonalization
Directions
: Please work on all of the problems and submit your solutions to the Calcula
tional Problems 1 and 2, and ProofWriting Problems 3 and 6 at the
beginning
of lecture
on March 9, 2007.
As usual, we are using
F
to denote either
R
or
C
. We also use
·
,
·
to denote an arbitrary
inner product and
·
to denote its associated norm. The term “o.n.” means
orthonormal
.
A
∗
denotes the conjugate transpose of a matrix
A
∈
C
n
×
n
.
1. Consider
R
3
with two orthonormal bases: the canonical basis
e
= (
e
1
, e
2
, e
3
) and the
basis
f
= (
f
1
, f
2
, f
3
), where
f
1
=
1
√
3
(1
,
1
,
1)
, f
2
=
1
√
6
(1
,
−
2
,
1)
, f
3
=
1
√
2
(1
,
0
,
−
1)
.
(a) Find the matrix,
S
, of the change of basis transformation such that
[
v
]
f
=
S
[
v
]
e
,
for all
v
∈
R
3
,
where [
v
]
b
denotes the column vector with the coordinates of the vector
v
in the
basis
b
.
(b) Find the canonical matrix,
A
, of the linear map
T
∈ L
(
R
3
) with eigenvectors
f
1
, f
2
, f
3
and eigenvalues 1
,
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 Fall '07
 Schilling
 Linear Algebra, Algebra, Hilbert space, F3, Hermitian, canonical basis

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