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
, give a basis for image(
T
).
Do
you have enough information to compute these vectors explicitly?
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 Fall '05
 HUI
 Math, Linear Algebra, Algebra, Vector Space, linear transformation, R10

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