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:

() − 21 + 2 − 3 = 4 () 1 − 22 + 33 = −21 + 22 + 33 = 13 − 1 + 2 − 23 = 3

31 + 3 = −1 21 − 2 + 33 = 1

() 1 − 22 + 33 = −2 () 71 + 22 − 23 − 44 + 35 = 8−1 + 2 − 23 = 3 − 31 − 32 + 24 + 5 = −121 − 2 + 33 = −7 41 − 2 − 83 + 205 = 1

2. Determine, where possible, the inverse of the following matrices:

() =

∙ −4 −25 5

¸() =

⎡⎣ 3 1 0

−1 2 2

5 0 −1

⎤⎦ () =

⎡⎣ 3 3 6

0 1 2

−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:

31 + 2 = 6

−1 + 22 + 23 = −751 − 3 = 10

4. Determine the conditions (if any) on 1 2 3 in order for the

following systems to be consistent:

() 1 − 22 + 63 = 1 () 1 + 32 − 23 = 1

−1 + 2 − 3 = 2 − 1 − 52 + 33 = 2

−31 + 2 + 83 = 3 21 − 82 + 33 = 3

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 = 31 − 42 1 = 422 + 23

2 = 1 + 22 2 = 21 − 23

3 = 61 − 2

4 = 102

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 + 21 2 + 2)

(d) The set = { ∈ | 0} with addition and scalarmultiplication 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

7. Determine if the given sets are subspaces of the respective vector

spaces:

() = {( ) ∈ 2 | ≥ 0};2() = {( ) ∈ 3 | = 0};3() = {( ) ∈ 3 | = 1};3In 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

−4 10

¸; =

∙1 −7−3 21

¸; =

∙0 0

0 0

¸

9. Describe the span of each of the following sets of “vectors”:

() 1 =

∙1 0

0 0

¸ 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

13. Determine if the following sets of vectors are linearly indepen-

dent or linearly dependent:

() 1 =

∙1 0 0

0 0 0

¸ 2 =

∙0 0 1

0 0 0

¸ 3 =

∙0 0 0

0 1 0

¸;

() 1 =

∙1 2

0 −1¸ 2 =

∙4 1

0 −3¸;

() 1 = 1 2 = 3 = 2 in 2;

() 1 = 22 − + 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 =

∙1 0

0 0

¸ 2 =

∙0 0

1 0

¸

3 =

∙0 1

0 0

¸ 4 =

∙0 0

0 1

¸

16. Determine the basis and dimension of the null space of the

following matrices:

() =

⎡⎣ 7 2 −2 −4 3

−3 −3 0 2 1

4 −1 −8 0 20

⎤⎦ ; () =⎡⎣ 2 −4 1 2 −2 −3−1 2 0 0 1 −110 −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

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 −82 0 −6 −2 4

−3 1 7 −2 12

⎤⎥⎥⎦ ; () =⎡⎣ 6 −3−2 3

−8 4

⎤⎦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

−1 −4¸; () =

∙7 −14 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

−1 −6 −25 0 0

⎤⎦ ; () =⎡⎣ 6 3 −80 −2 0

1 0 −3

⎤⎦ ;() =

⎡⎣ 0 1 1

1 0 1

1 1 0

⎤⎦ ; () =

⎡⎣ 4 0 −10 3 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) = 321 − 212 + 313 + 2 − 423 + 323

6

23. Classify the following quadratic forms, whether positive defi-

nite, positive semidefinite etc.:

() (1 2) = 21 + 822; () (1 2 3) = 22 + 8

23;

() (1 2 3) = 521 + 213 + 222 + 223 + 4

23;

() (1 2 3) = −321 + 212 − 22 + 423 − 823