3
18.06
Professor
Strang
Quiz
1
February
28,
2005
Grading
1
2
Your
PRINTED
name
is:
SOLUTIONS
4
1
(26
pts.)
Suppose
A
is
reduced
by
the
usual
row
operations
to
�
⎡
1
4
0
2
�
⎢
�
⎢
R
=
�
0
0
1
2
⎢
.
�
⎣
0
0
0
0
Find
the
complete
solution
(if
a
solution
exists)
to
this
system
involving
the
original
A
:
Ax
=
sum
of
the
columns
of
A.
Solution
The
complete
solution
x
=
x
particular
+
x
nullspace
has
� ⎡
�
⎡
�
⎡
1
−
4
−
2
� ⎢
�
⎢
�
⎢
� ⎢
�
⎢
�
⎢
�
1
⎢
�
1
⎢
�
0
⎢
� ⎢
�
⎢
�
⎢
=
.
x
particular
� ⎢
x
nullspace
=
x
2
�
⎢
+
x
4
�
⎢
�
1
⎢
�
0
⎢
�
−
2
⎢
� ⎣
�
⎣
�
⎣
1
0
1
The
free
variables
x
2
and
x
4
can
take
any
values.
The
two
special
solutions
came
from
the
nullspace
of
R
=
nullspace
of
A
.
The
particular
solution
of
1’s
gives
Ax
=
sum
of
the
columns
of
A
.
Note
:
This
also
gives
Rx
=
sum
of
columns
of
R
.
1
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2
(18
pts.)
Suppose
the
4
by
4
matrices
A
and
B
have
the
same
column
space
.
They
may
not
have
the
same
columns
!
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 Spring '05
 Strang
 Linear Algebra, column space

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