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18.06  Final
Exam,
Monday
May
16th,
2005
solutions
1.
(a)
We
want
the
coordinates
(
a
i
1
,...,a
in
)
to
satisfy
the
equation
c
1
x
1
+
...
+
c
n
x
n
=1;
thus
the
system
of
equations
we
want
to
solve
is
Ac
=
ones
:
c
1
a
11
+
c
2
a
12
+
+
c
n
a
1
n
=1
c
1
a
21
+
c
2
a
22
+
+
c
n
a
2
n
c
1
a
n
1
+
c
2
a
n
2
+
+
c
n
a
nn
(b)
There
is
no
plane
of
the
given
form,
if
one
of
the
points
P
i
is
the
origin.
In
R
3
an
example
is
given
by
the
points
(0
,
0
,
0),
(1
,
0
,
0)
and
(0
,
1
,
0):
they
lie
in
the
(unique)
plane
x
3
=0
and
this
plane
does
not
have
the
required
form.
More
than
one
plane
contains
the
P
i
’s
if
the
three
points
are
on
a
line
not
through
the
origin.
(c)
There
is
not
a
unique
solution
precisely
det
A
=
0.
This
means
geometrically
that
the
points
P
i
lie
in
an
(
n
−
1)
−
dimensional
subspace
of
R
n
.
2.
(a)
Subtracting
the
ﬁrst
row
from
the
second,
we
ﬁnd
the
matrix
⎡
⎤
12345
⎦
U
=
⎣
00001
.
00000
(In
the
row
reduced
echelon
form
R
,
the
5
changes
to
0.)
The
pivot
variables
are
the
ﬁrst
and
the
last,
while
the
remaining
ones
are
the
free
variables.
Thus
the
“special
solutions”
to
Ax
are
⎡
⎤
⎡
⎤
⎡
⎤
−
2
−
3
−
4
⎢
1
⎥
⎥
⎢
0
⎥
⎢
0
⎢
⎥
⎢
⎥
⎢
⎥
⎥
⎥
⎥
⎢
0
,
⎢
1
,
⎢
0
.
⎢
⎥
⎢
⎥
⎢
⎥
⎣
0
⎦
⎦
⎣
1
⎦
⎣
0
0
0
0
(b)
and
(c)
need
to
prove
that
these
three
vectors
are
linearly independent
and
they
span the nullspace
.
By
considering
the
second,
third
and
fourth
coordinates,
a
combination
of
the
vectors
adding
to
zero
must
have
zero
coeﬃcients.
The
vectors
span
the
nullspace,
since
the
dimension
of
the
nullspace
is
three
(note
that
the
rank
of
the
matrix
A
is
2).
3.
(a)
The
condition
that
Ax
=
b
has
no
solution
means
that
the
column
space
of
A
has
dimension
strictly
smaller
than
m
.
In
particular,
the
rank
is
r<
m
.S
i
n
c
e
A
T
y
=
c
has
exactly
one
solution,
the
columns
of
A
T
are
independent.
This
means
that
the
rank
of
A
T
is
r
=
m
.
This
contradiction
proves
that
we
cannot
ﬁnd
A
,
b
and
c
.
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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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