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18.06
Professor Strang
Final
Exam
May
16,
2005
Grading
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PRINTED
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Closed
book
/
10
wonderful
problems.
Thank
you
for
taking
this
course !
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is
an
extra
blank
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at
the
end.
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View Full Document 1(
1
0
p
t
s
.
)
Suppose
P
1
,...,P
n
are
points
in
R
n
.
The
coordinates
of
P
i
are
(
a
i
1
,a
i
2
,...,a
in
).
We
want
to
ﬁnd
a
hyperplane
c
1
x
1
+
···
+
c
n
x
n
=
1
that
contains
all
n
points
P
i
.
(a)
What
system
of
equations
would
you
solve
to
ﬁnd
the
c
’s
for
that
hyperplane
?
(b)
Give
an
example
in
R
3
where
no
such
hyperplane
exists
(of
this
form),
and
an
example
which
allows
more
than
one
hyperplane
of
this
form.
(c)
Under
what
conditions
on
the
points
or
their
coordinates
is
there
not
a
unique
interpolating
hyperplane
with
this
equation
?
2
2(
1
0
p
t
s
.
)
(a)
Find
a
complete
set
of
“special
solutions”
to
Ax
=
0
by
noticing
the
pivot
variables
and
free
variables
(those
have
values
1
or
0).
⎡
⎤
12345
⎢
⎥
⎢
⎥
A
=
⎢
12346
⎥
.
⎣
⎦
00000
(b)
and
(c)
Prove
that
those
special
solutions
are
a
basis
for
the
nullspace
N
(
A
).
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This note was uploaded on 01/14/2011 for the course EECS 18.06 taught by Professor Strang during the Spring '05 term at University of Michigan.
 Spring '05
 Strang

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