Newberger Math 247 Spring 03
Homework solutions: Section 1.4, #3136
31. Let
A
be a 3
×
2 matrix. Explain why the equation
A
x
=
b
cannot
be consistent for all
b
in
R
3
. Generalize your argument to the case of
an arbitrary
A
with more rows than columns.
Notice that saying the equation
A
x
=
b
is consistent for all
b
in
R
3
is the same thing as saying that the columns of
A
span all of
R
3
.
Theorem 4 says that the columns of
A
span all of
R
3
only when
A
has
a pivot position in every row. Since
A
is
3
×
2
there can be at most
2
pivots, so there cannot be a pivot in every row.
Geometrically, we can see that the largest subset of
R
3
that the
2
columns of
A
can span is a plane, so there will be some vectors
b
that
will not lie in this plane, and the system
A
x
=
b
will not always be
consistent.
Generally, if
A
has more rows than columns, there cannot be a pivot
in every row, and the system
A
x
=
b
will not always be consistent.
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 Spring '03
 F,newberger
 Math, ax, pivot positions

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