(a) What is
n
?
(b) What is
m
?

(c) Are
u
, . . . , u
linearly independent?

1
m
(d) Does
{
u
, . . . , u
}
span
R
n
?

1
m
(e) Looking at
B
can you write down a subset of the original set
{
u
, . . . , u
}
that

1
m
would be guaranteed to be linearly independent?
(f) Is there a subset of the original set
{
u
, . . . , u
}
that would be guaranteed to

1
m
span
R
n
?

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Chapter 10 / Exercise 1

**Mathematical Applications for the Management, Life, and Social Sciences**

Harshbarger

Expert Verified

(g) Write down a
b
∈
R
n
for which
Bx
=
b
does not have a solution.
(h) Write down a
b
∈
R
n
for which
Bx
=
b
has a solution.
(i) Write down a
b
∈
R
n
for which
Bx
=
b
has a unique solution.
(j) Is there a new vector
w
∈
R
n
that you could add to the set
{
u
1
, . . . , u
m
}
to
guarantee that
{
u
1
, . . . , u
m
, w
}
will span
R
n
?
(k) Is there a column of
B
that is in the span of the rest? If so, find it.
(l) Looking at
B
do you see a
u
i
that is in the span of the others? How can you
identify it?
(m) Assuming that no row of
A
was a zero row, how many planes are being used to
cut out the solutions of
Ax
= 0?