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Prelim 1
Math 294
September 28, 2006
4 questions; total 100 points
. You may use anything that has been given
in class or in the book, as long as you
show clearly what you are using
.
Calculators are neither needed nor permitted. Some pieces of formulas and
deﬁnitions can be found on the bottom of the second side.
1. We are given a system of 4 linear equations in 6 unknowns in the form
A~x
=
~
b
, and told that the augmented matrix
A
~
b
row reduces to
1
3
0
1
1
0
1
0
0
1
1
2
0
2
0
0
0
0
0
1
3
0
0
0
0
0
0
0
(a)
(10 points)
Find all solutions to
A~x
=
~
b
.
(b)
(4 points)
Let
T
:
R
p
→
R
q
be the linear transformation given by
T
(
~x
) =
A~x
. Determine
p
and
q
.
(c)
(4 points)
Determine rank(
A
) and the dimension of image(
T
).
(d)
(4 points)
Is it true that
~
b
is an element of image(
T
)? Why or
why not?
(e)
(4 points)
Determine the dimension and a basis for the kernel of
T
.
(f)
(4 points)
If
A
=
±
~v
1

~v
2
···
~v
6
²
, give a basis for image(
T
). Do
you have enough information to compute these vectors explicitly?
How would you do it, if so? (Do not do the calculations.)
2. Consider the matrix
A
=
1 2 0
0 1 2
1 2 1
.
(a)
(15 points)
Find
A

1
and check your answer using matrix multi
plication.
(b)
(5 points)
If
T
is the linear transformation deﬁned by
T
(
~x
) =
A~x
then determine the vector
~v
so that
T
(
~v
) =
1
1
1
.
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View Full Document3. Let
T
be the linear transformation that reﬂects
R
3
through the plane
x

y
+ 2
z
= 0.
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B
B
B
B
B
B
T
(
~x
)
~x
(a)
(10 points)
Determine a basis
B
=
{
~v
1
,~v
2
,~v
3
}
for
R
3
so that
~v
1
is
orthogonal to the plane and
~v
2
,~v
3
are on the plane.
(b)
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 Spring '05
 HUI
 Math, Linear Algebra, Algebra

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