Download University of Bahrain

University of Bahrain
College of Science
Department of Mathematics
Summer Semester 2008/2009
Final Examination
MATHS 211
Date : 24th August 2009
Max Marks: 50
Time: 08:30-10:30
Name:
I.D. No:
Marking Scheme
Question
Max. Marks
1
10
2
12
3
20
4
8
Total
50
Marks Obtained
Good Luck
1
Section:
QUESTION 1 [ 4,6 marks]
a)
Find the eigenvalues and basis of eigenspaces of
3  1
A

4  1
Is A diagonalizable?
2
b)
Let T : M 22  P2 be linear transformation defined as
 a b  
  a  2b  c   a  c  x  a  d  x 2
T  


 c d  
Find (i) a basis of kernel T.
(ii) a basis of Range T.
3
QUESTION 2 [ 12 marks]
a) Show whether the set A, B, C is linearly dependent or linearly independent in M 22 , where
1  2
0 1 
1  1
A
,B
,C  



0  1 
3 4
0 2 
4
b) Find the matrix X if
2 1 
  1 2 1 0
 3 4 X   4 0 3 1 




c) If A is n  n matrix and det A  0
show whether the rows of A forms a linearly dependent or linearly independent set.
5
QUESTION 3 [20 marks]
a) If T : P2  P3 is a linear transformation such that
T ( x 2 )  x 3 , T x  1  1, T x  1  x
Find T ( x 2  4 x  2)
1 1
b) Let W  X in M 22 : AX  X  where A  

0 0 
Find a basis of W.
6
c) For the system
x1  2 x2  x3  b1
 x1  3x2  x3  b2
 2 x1  4 x2  2 x3  b3
Find for what relation between b1 , b2 and b3 the system has no solution.

d) Show that C 1  D 1

1
 C (C  D) 1 D
7
e) If A is 2  2 matrix, the eigenvalues of A-1 are   3 and   2 , the basis of
 4  
1 
eigenspaces for   3 is    and for   2 is   
1 
1 
2
Find the matrix A .
8
QUESTION 4 [8 marks]
In each part one of the 4 statements is correct, circle the correct statement
a) (i) If A is diagonalizable then A-1 must exist.
(ii) If A is m n m  n then the rows or columns of A must be linearly dependent.
(iii) If A is 5 4 then the system AX= 0 always have infinite solutions.
(iv) If a matrix A is a square then rank A = nullity A.
b) If S is a basis of V and S has n vectors then
(i) any set of less than n vectors in V is always linearly dependent
(ii) Any set of n vectors is also a basis of V.
(iii) A vector u s is a vector in R n
(iv) A set v1 , v2 , u does not span V for any vector u in V.
c) If A is n  n matrix then
(i) A diagonalizable only if A has n different eigenvalues.
(ii) If   0 is an eigenvalue of A then A is not singular.
(iii) The system AX  B, always have no solution if A-1 does not exist.
(iv) If   c is an eigenvalue of A then   c 2  2c is an eigenvalue of A2  2 A .
9
d) Let S  v1 , v2 , v1   2,1, v2  2,1
(i) S is a basis of R 2 .
(ii) W= span S, has dimension = 1.
(iii) If the vector u is not in span S then v1 , v2 , u is linearly independent set.
(iv) The vector (0,0) is not in span S.
10