huynh (lth436) – HW08 – gilbert – (57245)
1
This
print-out
should
have
20
questions.
Multiple-choice questions may continue on
the next column or page – find all choices
before answering.
001
10.0points
Four linearly independent vectors span
R
5
.
True or False?
1.
TRUE
2.
FALSE
correct
Explanation:
The space
R
5
is 5-dimensional, so five lin-
early independent vectors are needed to form
a basis, and hence span
R
5
.
Consequently, the statement is
FALSE
.
002
10.0points
The pivot columns of rref(
A
) form a basis
for Col
A
.
True or False?
Let
H
be the set of all vectors
a
−
2
b
ab
+ 3
a
b
where
a
and
b
are real.
Determine if
H
is
a subspace of
R
3
, and then check the correct
answer below.
1.
H
isnotasubspaceof
R
3
becauseitdoes
notcontain
0
.
1.
TRUE
2.
FALSE
correct
Explanation:
The columns of
A
have the same linear de-
pendence relation as the columns of rref(
A
).
Also, the pivot columns of
A
are linearly in-
dependent and span Col
A
, hence form a basis
for Col
A
. But the columns of rref(
A
) may not
span Col
A
, so the pivot columns of rref(
A
)
need not form a basis for Col
A
.
For example, when
A
=
1
2
3
2
5
8
2
4
6
,
then
rref(
A
) =
1
0
−
1
0
1
2
0
0
0
.
2.
H
is a subspace of
R
3
because it can be
writtenasNul
(
A
)
forsomematrix
A
.
3.
H
is a subspace of
R
3
because it can be
writtenasSpan
{
v
1
,
v
2
}
with
v
1
,
v
2
in
R
3
.
4.
H
is not a subspace of
R
3
because it is
notclosedundervectoraddition
.
correct
Explanation:
To check if the set
H
of all vectors
a
−
2
b
ab
+ 3
a
b
is a subspace of
R
3
we check the properties
defining a subspace:
Thus the first two columns of
A
form a ba-
sis for Col
A
because the first two columns of
rref(
A
) are pivot columns.
But the last en-
try is 0 in each pivot column of rref(
A
), so the
columns of
A
can never be a linear combina-
tion of the pivot columns of rref(
A
).
Consequently, the statement is
FALSE
.
003
10.0points

huynh (lth436) – HW08 – gilbert – (57245)
2
1.
the zerovector
0
isin
H
: set
a
=
b
= 0.
Then
0
−
0
0 + 0
0
=
0
0
0
,
so
H
contains
0
.