MAS 3105
AM: Feb 28, 2008
Quiz 4 and Key
Prof. S. Hudson
1) [40 pts] Set
B
=
{
v
1
,
v
2
,
v
3
}
=
{
(1
,
0
,
0)
T
,
(1
,
1
,
0)
T
,
(1
,
0
,
1)
T
}
. So,
B
is a set of 3
column vectors in
R
3
. Answer, and explain brieﬂy:
a) Is
B
a basis of
R
3
?
b) Find the transition matrix from
B
to the standard basis of
R
3
.
c) Notice that
w
= (3
,
1
,
0)
T
is a simple LC of
v
1
and
v
2
. Find the coordinates of
w
with
respect to
B
. Write your answer as a column vector.
d) Suppose
B
2
=
{
v
3
,
v
2
,
v
1
}
. Find the transition matrix from
B
2
to
B
.
e) Find the coordinates of
w
with respect to
B
2
(as in part c). Check your answer to (d)
using your answers to (c) and (e).
2) Choose ONE of these to prove. You can answer on the back.
a) If the columns of
A
are LI and
Ax
=
b
is consistent, then the system has a
unique
solution. This is a rephrasing of Thm 3.3.2; so, prove it, don’t just quote it.
b) If dim(V)=
n >
0 and
B
=
{
v
1
,v
2
,...,v
n
}
spans
V
, then
B
is a basis of V.
c) Any two bases of