ma253
,Fa
l
l
2007
— Problem Set
5
This problem set deals mostly with subspaces, spans, linear dependence
and independence, and bases. As usual, solve all the problems, then write up
and turn in those marked with an asterisk. Given Fall Break, this assignment
is due on
Friday, October 26
.
1.
Problems from the textbook:
a. Section
3
.
1
, problems *
18
,*
22
,*
38
,*
48
.
b. Section
3
.
2
, problems
1
,*
2
,
3
,*
4
,*
6
,
8
,
10
,
12
,*
14
,*
16
,*
20
,*
32
,
*
34
,*
36
,*
37
.
c. Section
4
.
1
, problems
1
,*
2
,
3
,*
4
,
5
,*
12
,
13
,
14
.
*2.
Suppose you have a linearly dependent set
{
~
v
1
,
~
v
2
,...,
~
v
k
}
. Show that
one of the
~
v
i
can be written as a linear combination of the others. (Yes, we
did this in class; write it up carefully.) Is it true that
any
of the
~
v
i
can be
written this way?
*3.
Let
A
be an
n
×
n
matrix. Show that the following are equivalent:
a.