MATH 110  PRACTICE FINAL
LECTURE 1, SUMMER 2009
August 6, 2009
As on the practice midterm, the ﬁnal will consist of much fewer problems than the number
given here — probably six. These problems range from easy to medium to “almost”hard, but
I don’t think any are as diﬃcult as the starred problems from the practice midterm.
All vector spaces can be assumed to be nonzero and ﬁnitedimensional over the ﬁeld of real
numbers or the ﬁeld of complex numbers. Have fun!
1.
Axler, 6.18 (Hint: By 6.17 it is enough to show that every vector in null
P
is orthogonal to
every vector in range
P
— use 6.2 to show this)
2.
Axler, 6.20
3.
Suppose that
V
is an innerproduct space and that
U
and
W
are subspaces of
V
. Show
that
U
⊆
W
if and only if
W
⊥
⊆
U
⊥
.
4.
Suppose that
S
and
T
are selfadjoint operators on an innerproduct space
V
. Show that
ST
is selfadjoint if and only if
S
and
T
commute.
5.
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 Spring '08
 GUREVITCH
 Math, Linear Algebra, Vector Space, Complex number, Hilbert space

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