MTE-02

ASSIGNMENT BOOKLETBachelor’s Degree Programme

(B.Sc./B.A./B.Com.)

LINEAR ALGEBRA

(1st January, 2016 to 31st December, 2016)

• It is compulsory to submit the Assignment before filling in theTerm-End Examination form.

• It is mandatory to register for a course before appearing in theTerm-End Examination of the course. Otherwise, your result willnot be declared.

For B.Sc. Students Only

• You can take electives(56 or 64 credits) from a minimum of TWOand a maximum of FOUR science disciplines, viz. Physics,Chemistry, Life Sciences and Mathematics.

• You can opt for elective courses worth a MINIMUM OF 8 CREDITSand a MAXIMUM OF 48 CREDITS from any of these four disciplines.

• At least 25% of the total credits that you register for in theelective courses from elective courses from Life Sciences,Chemistry and Physics discipline must be from the laboratorycourses. For example, if you opt for a total of 24 credits of electives in these 3 disciplines, at least 6 credits should be fromlab courses.

School of Sciences

Indira Gandhi National Open University

Maidan Garhi, New Delhi-110068

( 2016)

 

Dear Student,

Please read the section on assignments in the Programme Guide for elective Courses that we

sent you after your enrolment. A weightage of 30%, as you are aware, has been earmarked for

continuous evaluation,  which would consist of one tutor-marked assignment for this course.

The assignment is in this booklet.

Instructions for Formatting Your Assignments

Before attempting the assignment please read the following instructions carefully.

1) On top of the first page of your answer sheet, please write the details exactly in the

following format:

ROLL NO. : . . . . . . . . . . . . . . . . . . . . . . . . . .

NAME : . . . . . . . . . . . . . . . . . . . . . . . . . .

ADDRESS : . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . .

COURSE CODE :   . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

COURSE TITLE :   . . . . . . . . . . . . . . . . . . . . . . . .

STUDY CENTRE :  . . . . . . . . . . . . . . . . . . . . . . . .   DATE . . . . . . . . . . . . . . . . . . . . . . . . . .

PLEASE FOLLOW THE ABOVE FORMAT STRICTLY TO FACILITATE

EVALUATION AND TO AVOID DELAY.

2) Use only foolscap size writing paper (but not of very thin variety) for writing your answers.

3) Leave a 4 cm margin on the left, top and bottom of your answer sheet.

4) Your answers should be precise.

5) While solving problems, clearly indicate which part of which question is being solved.

6) This assignment is to be submitted to the Study Centre as per the schedule made by the

study centre.  Answer sheets received after the due date shall not be accepted.

7) This assignment is valid only up to 31st December, 2016. If you fail in this assignment or

fail to submit it by 31st December, 2016, then you need to get the assignment for the year

2017 and submit it as per the instructions given in the Programme Guide.

8)   You cannot fill the Exam form for this course till you have submitted this assignment. So

solve it and submit it to your study centre at the earliest.

9)   We strongly suggest that you retain a copy of your answer sheets.

We wish you good luck.

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Assignment

Course Code: MTE-02

Assignment Code: MTE-02/TMA/2016

Maximum Marks: 100

1) Which of the following statements are true and which are false? Justify your answer with a

short proof or a counterexample.

i) The set Q ×Z×R contains the element

32 ,3

√ 2

.

ii) The function   f  :  R →R defined by   f ( x) = x2−3 is onto.

iii) If the scalar product of  a, b ∈ R3 is 0, then a = 0 or  b = 0.

iv) If   S 1 ⊂ S 2 are subsets of a vector space and  S 1 is linearly independent, then  S 2 is also

linearly independent.

v) The set {( x1, x2, x3) | x1, x2, x3 ∈ R, x1 x2 = x3} is a subspace of  R3.

vi) If all the eigenvalues of a matrix is zero, the matrix is the zero matrix.

vii) If the minimal polynomial of a matrix has distinct roots, the characteristic

polynomial of that matrix also has distinct roots.

viii) No unitary matrix can be hermitian.

ix) For any invertible matrix A, ( At )−1 =

 A−1t 

.

x) There is no co-ordinate transformation that transforms the quadratic form

 x2 + y2− z2 to the quadratic form x2− y2− z2. (20)

2) a) Which of the following are binary operations. Justify your answer.

i) The operation · defined on  Q  by  a ·b = a(b−1).

ii) The operation · defined on [0,π ] by  x · y = cos xy.

Also, for those operations which are binary operations, check whether they are

associative and commutative. (4)

b) Find the radius and the center of the circular section of the sphere |r| = 4 cut off by the

plane r · (2i− j + 4k) = 3. (4)

c) If  V  and W  are subspaces of  R7 of dimensions 3 and 4, respectively, check that the

dimension of V ∩W   is 1, 2 or 3. (2)

3) Let P5 be the vector space over  R  defined by

P5 =

  5

∑i=0

ai xi

ai ∈ R

and let

W  =

  5

∑i=0

ai xi

a0 = a2 = a4 = 0,ai ∈ R

a) Check that W  is a subspace of  P5. What is the dimension of W ? (3)

b) If   p( x), q( x) ∈ P5, what is the condition for  p( x) +W  = q( x) +W ? (2)

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c) Find a subspace U  of  P5 such that W ⊕U  =  P5. What is the dimension of U ? (5)

4) a) Let P5 and W  as in question 3. Show that  P5

W   P2 by defining a surjective linear map

from P5 to  P2 with kernel W  and applying the fundamental theorem of 

homomorphism. (3)

b) Let V  be a vector space over a field  F  and let  T  :  V  →V  be a linear operator. Show

T (W ) ⊂W  for any subspace W  of  V  if and only if there is a  λ  ∈ F such that  T v =  λ v

for all v ∈V . (7)

5) a) Find the rank and nullity of the matrix

 A =

1 2 0 5

−1 3   −5 5

3 2 4 7

by row reduction. (3)

b) Let T  :  P1 → P2 be defined by

T (a + bx) = b + ax + (a−b) x2.

Check that T  is a linear transformation. Find the matrix of the transformation with

respect to the ordered bases  B1 = {1, x} and B2 = { x2, x2 + x, x2 + x + 1}. Find the

kernel of  T . Further check that the range of  T   is

{a + bx + cx2 ∈ P2 | a + c = b}.   (7)

6) a) Check whether the matrices A  and  B  are diagonalisable. Diagonalise those matrices

which are diagonalisable. (7)

i)   A =

0 3   −5

1   −1 4

1   −2 5

  ii)   B =

2   −2 6

−2 2   −6

−1 1   −3

.

b) Find the minimal polynomials of  A  and  B  in part a). (3)

7) a) Find the values of  a, b  and  c ∈ C for which the matrix

1   i   1 + 2i

a   0   b

c   2 + i   1

is Hermitian. (2)

b) Find the values of  a,b ∈ C for which the matrix

1 0 0

0   −1/√ 

2 1/√ 

2

0   a b

is unitary. Justify your answer. (5)

c) Let ( x1, x2, x3) and ( y1, y2, y3) represent the coordinates with respect to the bases

 B1 = {(1,0,0),(0,1,0),(0,0,1)}, B2 = {(1,0,0),(0,1,2),(0,2,1)}. If 

Q( X ) = 2 x21 + 2 x1 x2−2 x2 x3 + x2

2 + x23, find the representation of  Q  in terms of 

( y1, y2, y3). (3)

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8) a) If   x = ( x1, x2) and y = ( y1, y2) are in C2, define  x, y = 2 x1 y1−3 x2 y2. Which of the

properties for an inner product are satisfied and which are not satisfied by  x, y?Justify your answer. (4)

b) Apply the Gram-Schmidt diagonalisation process to find an orthonormal basis for the

subspace of  C4 generated by the vectors

{(1,−i,0,1), (−1,0, i,0),(−i,0,1,−1)}   (6)

c) Find the orthogonal canonical reduction of the quadratic form

− x2 + y2 + z2−6 xy−6 xz + 2 yz. Also, find its principal axes, rank and signature of the

quadratic form. (6)

d) Find the nature of the conic 2 x2 + xy− y2−6 = 0. (4)

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