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Worksheet 4
February 2nd, 2009
1. If
A
x
=
A
y
must
x
=
y
? If so explain why, if not, ﬁnd an
A
and two
vectors
x
6
=
y
such that
A
x
=
A
y
2. Construct two 3
×
3 matrices
A
and
B
such that
AB
= 0,
A
6
= 0,
B
6
= 0, and
BA
6
= 0.
3. Suppose for
T
:
R
n
→
R
n
that
T
(
x
) =
b
and
T
(
y
) =
b
for some
x
6
=
y
,
can
T
map
R
n
onto
R
n
.
4. We know we can write any function
f
:
R
2
→
R
2
as
f
(
x,y
) = (
f
1
(
x,y
)
,f
2
(
x,y
)).
Consider the linear transformation
f
(
x
) =
A
x
where
A
=
±
3
4
2

1
²
.
Write this function as
f
(
x,y
) = (
f
1
(
x,y
)
,f
2
(
x,y
)).
5. Matrix multiplication is not necessarily commutative.
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