Math 2360
Exam III
Form A
18 November 2009
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. (9 pts)
Determine whether the following are linear transformations from  4 to  3
2. (9 pts)
a.
 x1  x3

 x  2 x4 
 2

x2  x4 

L( x )   x1 
x3 


 x2  x4 




c.
1 
L( x )   0 
1 


 x1  2 x2  4 x4 


x2  x3  x4 

L( x )  x1 


3
x  x  x

 `1 2 3  x4 
3


b.
Determine whether the following are linear transformations from P4 to P4
p (0) 2
x
2
a.
L ( p ( x ))  p ( x ) 
c.
L ( p ( x ))  p (0) p ( x )
b.
L ( p ( x ))  p ( x )  xp ( x )
Section II
Continue the answers for these problems on the answer sheet. Append additional pages as needed.
You do not need to rewrite the problem statements on your answer sheets. Work carefully. Do your
own work. Show all relevant supporting steps!
3. (10 pts)
Consider the linear transformation mapping  to  2 given by
4
L ( x )  [ x1  3 x2  x3  3 x4 , x1  x2  x3  x4 ]T
a.
b.
4. (10 pts)
Find the kernel of L.
Find the dimension of the range of L.
Consider the linear transformation mapping P3 to P4 given by
x
L ( p ( x )) 
 p(t )dt
0
a.
b.
Find the kernel of L.
Find the dimension of the range of L.
5. (8 pts)
Consider the linear transformation mapping  4 to  3 given by
L ( x )  [2 x1  x2  3 x3  4 x4 , x1  3 x2  x3  2 x4 , 2 x2  x3  5 x4 ]T
Find the standard matrix representation for L.
6. (16 pts)
Consider the linear transformation mapping  to  given by
3
2
L ( x )  [ x1  2 x2  3 x3 , 2 x1  x2  x3 ]T
Find a matrix A which represents L with respect the standard basis [ e1 , e 2 , e3 ] in  and the
3
1 
0
2
ordered basis [ b1 , b2 ] in  where b1    and b2    .
2
 1
7. (10 pts)
Find the equation of the plane in  which is normal to the vector N  [1,  1,  3]T and which
3
passes through the point P0  [ 3, 2,  1]T .
8. (10 pts)
9. (10 pts)
Find the distance in  from the point P0  [ 1,  2, 1]T to the plane given by
4 x  2 y  4 z  10 .
3
Let S be the subspace in  4 which is spanned by the set
1,  2, 1, 2 , 1, 2, 2, 1  .
T
Find a basis for S  .
10. (10 pts)
1 1
 1
0 1 
1



Find the least squares solution x of the linear system
x 
 2 1
 1


 
1 2 
1
T
Name _________________________
Form A
Answers
1.
i. Yes
No
ii. Yes
No
iii. Yes
No
i. Yes
No
ii. Yes
No
iii. Yes
No
2.