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Quiz #5
1. Label the following statements as being true or false. Assume that
V
is a ﬁnitedimensional inner product space
(a) An inner product is linear in both components.
(Solution)
False.
h
x,cy
i
= ¯
c
h
x,y
i
.
(b) If
x
,
y
, and
z
are vectors in
V
such that
h
x,y
i
=
h
x,z
i
, then
y
=
z
.
(Solution)
False. Take
x
= 0.
(c) If
h
x,y
i
= 0 for all
x
in
V
, then
y
= 0.
(Solution)
True. Take
x
=
y
.
(d) Every ﬁnitedimensional inner product space has an orthonormal
basis.
(Solution)
True. Use GramSchmidt orthogonalization process
to get an orthogonal basis from the given basis. And then nor
malize it!
(e) The orthogonal complement of any set is a subspace.
(Solution)
True. easy to prove it. Try!
(f) Every orthogonal set is linearly independent.
(Solution)
False. Consider the zero vector
(g) Every orthonormal set is linearly independent.
(Solution)
True. Try to prove it!
(h) For every linear operator
T
on
V
and every basis
β
for
V
, we
have [
T
*
]
β
= ([
T
]
β
)
*
.
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 Spring '06
 Holtz
 Linear Algebra, Algebra

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