Change of Orthonormal Basis Worksheet
In the change of basis worksheet, we saw that a pair of bases (
v
1
,...,v
n
) and (
v
0
1
,...,v
0
n
) for
a vector space
V
could be related by a invertible change of basis matrix
P
deﬁned by
(
v
0
1
,...,v
0
n
) = (
v
1
,...,v
n
)
P .
The columns of
P
were calculated by computing the component of the vectors (
v
0
1
,...,v
0
n
)
in the basis (
v
1
,...,v
n
). Also if
M
is the matrix of
f
∈
end(
V
) in the basis (
v
1
,...,v
n
), its
matrix
M
0
in the basis (
v
0
1
,...,v
0
n
) is given by
M
0
=
P

1
MP .
If you are lucky enough to have orthonormal bases, life is easier.
.... For example, consider
C
3
with standard inner product
h
(
z
1
,z
2
,z
3
)
,
(
w
1
,w
2
,w
3
)
i
=
∑
3
i
=1
¯
w
i
z
i
. An orthonormal basis is
f
1
= (1
,i,
0)
/
√
2
,f
2
= (1
,

i,
0)
/
√
2
,f
3
= (0
,
0
,
1). To compute the components of a vector
v
in this basis, you only have to compute
h
v,f
1
i
,
h
v,f
2
i
,
h
v,f
3
i
.
Calculate the components
of the vector
v
= (1
,
1
,
1)
in the basis
(
f
1
,f
2
,f
3
)
using the inner product.
Now let
e
1
= (1
,
0
,
0)
,e
2
= (0
,
1
,
0)
,e
3
= (0
,
0
,
1) be the canonical (orthonormal) basis for
C
3
.
Calculate the change of basis matrix
P
from the basis
(
e
1
,e
2
,e
3
)
to
(
f
1
,f
2
,f
3
). Remember
that
P
†
is obtained from
P
by taking the transpose and complex conjugate.
Compute
P
†
and
P
†
P
. What can you say about
P
†
? EXPLAIN YOUR OBSERVATION!
1