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King Fahd University of Petroleum & Minerals
Department of Math & Stat
Math-280, Term-141
Class Test No. 4
11-12-2014
Name: ___________________________
ID #: ____________________
(Show all your work)


1) Given the map L : P3  P3 defined by L p  x   p  x   p  x 
a) Show that L is a linear transformation.
b) Find the kernel of L .
c) Find the Range of L .
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2  1
1
2) Let A  
and S  e1 , e2 , e3  and T e1 , e2
3 
2  1
R 3 and R2 , respectively.

 be the natural bases for
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3
2
a) Find the linear transformation L : R  R whose representation with respect to
S and T is A .
b) Let
 1 

S    0  ,
 1 
 
1 
1  ,
 
0 
0  
1   and
 
1  
 1 
T     ,
  3
 2 
 1  
 
Be ordered bases for R 3 and R2 , respectively. Determine the linear transformation
L : R 3  R 2 , whose representation with respect to S  and T  is A .
c)
 1  


Compute L   2   using L as determined in part (b).
 3  
 
6  4
3) Given the Matrix A  

 3 1
a) Find the characteristic polynomial of A .
b) Find the eigenvalues and the associated eigenvectors.
c) Compute A  5 A  6 I
2
d) Show that if   0 is an eigenvalue of AB then,   0 is also an eigenvalue of BA .
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0 2
4) Given the Matrix A   2 0

 2 2
2
2
0
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a) Find a Matrix X that can Diagonalize A . Also, find the Diagonal Matrix D and its
relation with X and A .
b) Show that if A is diagonalizable, then AT is also diagonalizable.