Newberger Math 247 Spring 03
Homework solutions: Section 2.8 #3136
31. Suppose
F
is a 5
×
5 matrix whose column space is not equal to
R
5
. What can you say about Nul
F
?
Start your explanations with the assumptions (the suppose part).
Since the column space of
F
is not equal to
R
5
,
F
does not have a
pivot in every row. Since
F
is square, we know
F
also does not have
a pivot in every column.
This means that the homogeneous equation
F
x
=
0
has nontrivial solutions. Since Nul
F
is the solution set to
the homogeneous equation, we can deduce that Nul
F
contains nonzero
vectors, or in other words, Nul
F
6
=
{
0
}
.
32. If
R
is a 6
×
6 matrix and Nul
R
is not the zero subspace, what
can you say about Col
R
?
Start your explanation with the assumptions (the if part).
Since
Nul
R
is not the zero subspace, we know that the homogeneous equation
R
x
=
0
has nontrivial solutions. This means that
R
does not have a
pivot in every column. Since
R
is square, this also means that
R
does
not have a pivot in every row, so the columns of
R
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 Spring '03
 F,newberger
 Math, Linear Algebra, Vector Space, Elementary algebra, Empty set, Mathematical terminology

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