Prelim I Solutions
1. We are given a system of 4 linear equations in 7 unknowns in the form
A~x
=
~
b
, and told
that the augmented matrix
A
~
b
has reduced row echelon form
1
2
0
0
2
0
4
3
0
0
0
1
3
0
1
2
0
0
0
0
0
1
1
1
0
0
0
0
0
0
0
0
1. Find all solutions to
A~x
=
~
b
.
Solution:
x
1
+ 2
x
2

2
x
5
+ 4
x
7
=
3
x
4
+ 3
x
5
+
x
7
=
2
x
6

x
7
=
1
Let
x
2
=
a, x
3
=
b, x
5
=
c, x
7
=
d
x
1
=
3

2
a
+ 2
c

4
d
x
4
=
2

3
c

d
x
6
=
1 +
d
2. Let
T
:
R
p
→
R
q
be the linear transformation given by
T
(
~x
) =
A~x
. Determine
p
and
q
.
Solution:
Since the matrix of the transformation is a 4
×
7
p
= 7
q
= 4
3. Determine rank(
A
) and the dimension of image(
T
).
Solution:
Rank of A is the number of leading 1’s in the matrix which is 3, same as
the dimension of the image space.
4. Is it true that
~
b
is an element of image(
T
)? Why or why not?
Solution:
Yes, since
~
b
is the output when the matrix of the transformation acts on
any solution of part(a),
~
b
is in the image of T.
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5. Determine the dimension and a basis for the kernel of
T
.
Solution:
By rank nullity, dimension of the basis would be 4. Solving for the system
x
1
+ 2
x
2

2
x
5
+ 4
x
7
=
0
x
4
+ 3
x
5
+
x
7
=
0
x
6

x
7
=
0
~x
=
a

2
1
0
0
0
0
0
+
b
0
0
1
0
0
0
0
+
c
2
0
0

3
1
0
0
+
d

4
0
0

1
0
1
1
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 Spring '06
 PANTANO
 Linear Algebra, Linear Equations, Equations, Multivariable Calculus, Vector Space, basis

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