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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
deFnitions 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)
±ind 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

2
 · · · 
6
²
, give a basis for image(
T
). Do
you have enough information to compute these vectors explicitly?
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This note was uploaded on 10/01/2008 for the course MATH 2940 taught by Professor Hui during the Fall '05 term at Cornell University (Engineering School).
 Fall '05
 HUI
 Math, Linear Algebra, Algebra

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