Math 2360
Exam III
Form A
30 July 2008
Section I
For each of the problems in the this section at least one of the choices is correct. For some of the
problems more than one of the choices is correct. Record your answers for problems in this section on
the answer sheet (last page).
1. (3 pts)
Let A be a 4 × 6 matrix. If the dimension of the null space of A is 3, then the rank of A is:
a.
e.
2. (9 pts)
3. (9 pts)
Section II
4. (15 pts)
1
b.
3
c.
Undeterminable from the given information
4
d.
6
Determine whether the following are linear transformations from ¡ 3 to ¡ 2
a.
 x1 + x3

−
x
2

L( x) =  2


0


c.
0
L( x) =  
0
b.
 x1x3 − 2x 2 
L( x) = 

 x1 + x3 − 2 x2 
Determine whether the following are linear transformations from P3 to P3
p ′′(0) 2
x
2
a.
L ( p( x )) = p ( x ) −
c.
L ( p( x )) = p ( x ) + p ( − x )
b.
L ( p( x )) = p ( x ) − x
Answer the problems in this section on separate paper. You do not need to rewrite the problem
statements on your answer sheets. Work carefully. Do your own work. Show all relevant supporting
steps!
 2 −1 0 1 
Consider the matrix A given by A =  − 1 1 2 − 1 . A straightforward reduction by elimination


 3 −1 2 1 
1 0 2 0 
shows that A is row equivalent to U where U =  0 1 4 − 1 .


 0 0 0 0 
a.
b.
Find a basis for the row space of A.
Find a basis for the column space of A.
c.
5. (6 pts)
Find a basis for the null space of A.
Consider the linear transformation mapping ¡ 3 to ¡ 3 given by
L ( x ) = (2 x2 − x1 , x1 + 3 x2 − x3 , 2 x 3 − x 2 ) T
Find the standard matrix representation for L.
6. (10 pts)
Consider the linear transformation mapping ¡ 3 to ¡ 2 given by
L ( x ) = ( x1 + 3 x2 − x3 , x1 − 2 x2 )T
a.
b.
7. (10 pts)
Find the kernel of L.
Find the dimension of the range of L.
Consider the linear transformation mapping P3 to P3 given by
L ( p( x )) = ( p (1) − p ( −1)) x
a.
b.
8. (10 pts)
Find the kernel of L.
Find the dimension of the range of L.
Consider the linear transformation mapping P3 to P3 which satisfies the conditions that
L (1 + x ) = 1 + 2 x + 3 x 2
L (1 − x ) = 1 + 2 x
L (1 + x 2 ) = 1
Find the value of L (9 − 6 x + 3 x 2 )
9. (12 pts)
Consider the linear transformation mapping ¡ 3 to ¡ 2 given by
L ( x ) = ( x1 + 3 x2 − x3 , x1 − 2 x2 )T
Find a matrix A which represents L with respect the standard basis [ e1 , e2 , e3 ] in ¡ 3 and the ordered
1
2
2
0
basis [ b1 ,b2 ] in ¡ 2 where b1 =   and b2 =   .
10. (16 pts)
Consider the linear operator on ¡ 2 given by
L ( x ) = ( x1 − x2 , x1 − 2 x 2 ) T
Find a matrix A which represents L with respect to the standard basis [ e1 ,e 2 ] .
Find a matrix B which represents L with respect the ordered basis [ b1 ,b2 ] where
1
1
b1 =   and b2 =   .
1
 −1
Name _________________________
Form A
Answers
1.
______
2.
i. Yes
No
ii. Yes
No
iii. Yes
No
i. Yes
No
ii. Yes
No
iii. Yes
No
3.