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Math 136 Assignment #4
1. Let
A
be an
m
×
n
matrix and
~
b
∈
R
m
. Let
T
:
R
n
→
R
m
be the transformation deﬁned by
T
(
~x
) =
A~x
+
~
b
. Show that
T
is a linear transformation if and only if
~
b
=
~
0. (The transformation
of the above form is called an “aﬃne” transformation.)
2. Let
A
=
1
3 9
2
1
0 3

4
0
1 2
3

2 3 0
5
and
~
b
=

1
3

1
4
. Is
~
b
in the range of the linear transformation
~x
7→
A~x
? Explain your answer.
3. Let
T
:
R
n
→
R
m
be a linear transformation. Show that if
T
maps two linearly independent
vectors onto a linearly dependent set, then the equation
T
(
~x
) =
~
0 has a nontrivial solution.
4. Determine whether the following transformations are linear or not. If a transformation is
linear, then ﬁnd the corresponding standard matrix and check that the transformation is onto
or onetoone.
(a)
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This note was uploaded on 07/01/2008 for the course MATH 136 taught by Professor All during the Spring '08 term at Waterloo.
 Spring '08
 All
 Math, Linear Algebra, Algebra

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