south asian university assignment MA 1ST SEMESTER

1. 1
South Asian University
Quantitative Methods
Problem Set 3
Lecturer: Manimay Sengupta
Monsoon Semester, 2012.
1. Use Gauss-Jordan elimination procedure to solve the following
systems of linear equations:
() − 21 + 2 − 3 = 4 () 1 − 22 + 33 = −2
1 + 22 + 33 = 13 − 1 + 2 − 23 = 3
31 + 3 = −1 21 − 2 + 33 = 1
() 1 − 22 + 33 = −2 () 71 + 22 − 23 − 44 + 35 = 8
−1 + 2 − 23 = 3 − 31 − 32 + 24 + 5 = −1
21 − 2 + 33 = −7 41 − 2 − 83 + 205 = 1
2. Determine, where possible, the inverse of the following matrices:
⎡ ⎤ ⎡ ⎤
∙ ¸ 3 1 0 3 3 6
−4 −2
()  = ()  = ⎣ −1 2 2 ⎦ ()  = ⎣ 0 1 2 ⎦
5 5
5 0 −1 −2 0 0
⎡ ⎤
∙ ¸ 3 0 0
 
()  = ()  = ⎣ 0 2 0 ⎦
 
9 5 4
3. Use the inverse of the coefficient matrix to solve the following
system of equations:
31 + 2 = 6
−1 + 22 + 23 = −7
51 − 3 = 10
4. Determine the conditions (if any) on 1  2  3 in order for the
following systems to be consistent:
() 1 − 22 + 63 = 1 () 1 + 32 − 23 = 1
−1 + 2 − 3 = 2 − 1 − 52 + 33 = 2
−31 + 2 + 83 = 3 21 − 82 + 33 = 3

2. 2
5. Determine if the following functions are linear transformations
or not:
()  : 2 → 4 ()  : 3 → 2
 (1  2 ) = (1  2  3  4 )  (1  2  3 ) = (1  2 )
 1 = 31 − 42  1 = 42 + 2 2 3
2
2 = 1 + 22 2 = 1 − 2 3 
3 = 61 − 2
4 = 102 
6. Examine whether the following sets are vector spaces or not:
() The set  = 2 with the usual definition of vector addition,
and scalar multiplication defined as:
(1  2 ) = (1  2 )
() The set The set  = 3 with the usual definition of vector
addition, and scalar multiplication defined as:
(1  2  3 ) = (0 0 3 )
() The set  = 2 with the usual definition of scalar multiplica-
tion, and vector addition defined as:
(1  2 ) + (1  2 ) = (1 + 21  2 + 2 )
(d) The set  = { ∈  |   0} with addition and scalar
multiplication defined as follows:
 +  = ;  =  
(e) The set  of the points on a line through the origin in 2 with
the usual addition and scalar multiplication.
(f ) The set  of the points on a line that does not go through the
origin in 2 with the usual addition and scalar multiplication.
(g) The set  of the points on a plane that goes through the origin
in 3  with the standard addition and scalar multiplication.
(Note: The equation of a plane through the origin is + + =
0 where    are given constants.)

3. 3
7. Determine if the given sets are subspaces of the respective vector
spaces:
()  = {( ) ∈ 2 |  ≥ 0}; 2 
()  = {(  ) ∈ 3 |  = 0}; 3 
()  = {(  ) ∈ 3 |  = 1}; 3 
In what follows,  denotes the set of all  ×  matrices.
()  = All diagonal matrices of order ;  
()  = All 3 × 2 matrices such that 11 = 0; 32 
()  = [] , the set of all continuous functions  : [ ] →
; <[]  the set of all real-values functions defined on [ ],
()  =   the set of all polynomials of degree  or less; < the
set of all real-valued functions  :  → 
0
()  =   the set of all polynomials of degree exactly ; <
8. Determine the null space of each of¸the following matrices:
∙ ¸ ∙ ∙ ¸
2 0 1 −7 0 0
()  = ;= ;= 
−4 10 −3 21 0 0
9. Describe the span of each of the following sets of “vectors”:
∙ ¸ ∙ ¸
1 0 0 0
() 1 =  2 = ;
0 0 0 1
() 1 = 1 2 =  3 = 3 
10. Specify a set of vectors that will exactly span each of the
following vector spaces and verify your answers::
()  ; () 22 ; ()  
11. Verify if the following sets of vectors will span 3 :
() 1 = (1 2 0) 2 = (3 1 0) 3 = (4 0 1);
() 1 = (4 −3 9) 2 = (2 −1 8) 3 = (6 −5 10)
12. Determine if the following sets of vectors are linearly indepen-
dent or linearly dependent:
() 1 = (−2 1) 2 = (−1 −3) 3 = (4 −2);
() 1 = (1 1 −1 2) 2 = (2 −2 0 2) 3 = (2 −8 3 −1);
() 1 = (1 −2 3 −4) 2 = (−1 3 4 2) 3 = (1 1 −2 −2)

4. 4
13. Determine if the following sets of vectors are linearly indepen-
dent or linearly dependent:
∙ ¸ ∙ ¸ ∙ ¸
1 0 0 0 0 1 0 0 0
() 1 =  2 =  3 = ;
0 0 0 0 0 0 0 1 0
∙ ¸ ∙ ¸
1 2 4 1
() 1 =  2 = ;
0 −1 0 −3
() 1 = 1 2 =  3 = 2 in 2 ;
() 1 = 22 −  + 7 2 = 2 + 4 + 2 3 = 2 − 2 + 4 in 2 
14. Examine if each of the following sets of vectors will be a basis
for 3 :
() 1 = (1 0 0) 2 = (0 1 0) 3 = (0 0 1);
() 1 = (1 −2 1) 2 = (2 −1 3) 3 = (5 −3 −1);
() 1 = (1 1 0) 2 = (−1 0 0)
15. Examine if each of the following sets of vectors will form a
basis for the indicated vector space:
() 0 = 1 2 =  3 = 2    =  ;  
∙ ¸ ∙ ¸
1 0 0 0
() 1 =  2 = 
0 0 1 0
∙ ¸ ∙ ¸
0 1 0 0
3 =  4 = 
0 0 0 1
16. Determine the basis and dimension of the null space of the
following matrices:
⎡ ⎤ ⎡ ⎤
7 2 −2 −4 3 2 −4 1 2 −2 −3
()  = ⎣ −3 −3 0 2 1 ⎦ ; ()  = ⎣ −1 2 0 0 1 −1 ⎦ 
4 −1 −8 0 20 10 −4 −2 4 −2 4
17. Find the row and the column spaces of the following matrix:
⎡ ⎤
1 5 −2 3 5
⎢ 0 0 1 −1 0 ⎥
=⎢ ⎣ 0 0 0
⎥
0 1 ⎦
0 0 0 0 0

5. 5
18. Find a basis for the row and the column spaces of the matrices
 and  in Problem 16, and thus determine the rank of these matrices.
19. Find a basis for the row space, the column space and the null
space of the following matrices. Determine the rank and nullity of the
matrices.
⎡ ⎤
−1 2 −1 5 6 ⎡ ⎤
⎢ 4 −4 −4 −12 −8 ⎥ 6 −3
()  = ⎢⎣ 2
⎥ ; ()  = ⎣ −2 3 ⎦
0 −6 −2 4 ⎦
−8 4
−3 1 7 −2 12
20. For each of the following matrices, determine the eigenvectors
and a basis for the eigenspace of these matrices corresponding to each
of their eigenvalues:
∙ ¸ ∙ ¸
6 16 7 −1
()  = ; ()  = 
−1 −4 4 3
21. For each of the following matrices, determine the eigenvectors
and a basis for the eigenspace of these matrices corresponding to each
of their eigenvalues:
⎡ ⎤ ⎡ ⎤
4 0 1 6 3 −8
()  = ⎣ −1 −6 −2 ⎦ ; ()  = ⎣ 0 −2 0 ⎦ ;
5 0 0 1 0 −3
⎡ ⎤ ⎡ ⎤
0 1 1 4 0 −1
()  = ⎣ 1 0 1 ⎦ ; ()  = ⎣ 0 3 0 ⎦
1 1 0 1 0 2
22. Write the following quadratic forms in matrix form () =
0
  where  is a symmetric matrix:
(a) ( ) = 2 + 2 +  2 ; () ( ) = 2 +  +  2 ;
() (1  2  3 ) = 32 − 21 2 + 31 3 + 2 − 42 3 + 32 
1 3

6. 6
23. Classify the following quadratic forms, whether positive defi-
nite, positive semidefinite etc.:
() (1  2 ) = 2 + 82 ; () (1  2  3 ) = 2 + 82 ;
1 2 2 3
() (1  2  3 ) = 52 + 21 3 + 22 + 22 3 + 42 ;
1 2 3
() (1  2  3 ) = −31 + 21 2 − 2 + 42 3 − 82
2 2
3