ASSIGNMENT BOOKLET. Bachelor's Degree Programme LINEAR ALGEBRA. It is compulsory to submit the assignment before filling in the exam form.

Transcription

1 ASSIGNMENT BOOKLET MTE- Bachelor's Degree Programme LINEAR ALGEBRA (Valid from st January, to st December, ) It is compulsory to submit the assignment before filling in the exam form. School of Sciences Indira Gandhi National Open University Maidan Garhi, New Delhi-68 (For January, Cycle)

2 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 per cent, 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 Formating Your Assignments Before attempting the assignment please read the following instructions carefully. ) 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 :. ASSIGNMENT NO.:. STUDY CENTRE:.... DATE:.... PLEASE FOLLOW THE ABOVE FORMAT STRICTLY TO FACILITATE EVALUATION AND TO AVOID DELAY. ) Use only foolscap size writing paper (but not of very thin variety) for writing your answers. ) Leave cm margin on the left, top and bottom of your answer sheet. ) Your answers should be precise. 5) While solving problems, clearly indicate which part of which question is being solved. 6) This assignment responses are 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 upto December,. If you have failed in this assignment or fail to submit it by December,, then you need to get the next year s assignment and submit it as per the instructions given in the programme guide. Please retain a copy of your answer sheets. Wish you luck!

3 Assignment (To be done after studying the course material). Which of the following statements are true? Give reasons for your answers. Course Code: MTE- Assignment Code: MTE-/TMA/ Maximum Marks: i) Different eigenvectors corresponding to an eigenvalue of a matrix must be linearly dependent. ii) If A M n ( ) such that A, then A is diagonalisable. iii) If U and W are -dimensional subspaces of a 5-dimensional vector space V, then U W has at least one non-zero vector. 5x iv) The domain of f, defined by f (x) =, is { x x< }. x+ v) The range of any linear transformation from to is n m. vi) The matrices A and ka, k, k, have the same minimal polynomial, where A is a square matrix. vii) If { v, v, v} is a set of mutually orthogonal vectors, then so is { v + v, v + v, v + v}. viii) Given any n N, it is possible to define a linear transformation whose kernel has dimension n. x ix) The dimension of the solution set of the linear system given by [ a ij] n n = is the rank x n of [ a ij ]. x) If λ and µ are eigenvalues of two n n matrices A and B respectively, then µλ is an eigenvalue of AB. (). a) Give a relation on N which is reflexive and transitive but not symmetric. () b) If x is an odd integer, then prove that x (mod8). () c) Geometrically present the vectors u = i j, v= i+ j, u+ v, u v in a single diagram. () d) Find the radius of the circular section of the sphere r c = 7 by the plane r. ( i j+ k) = 7, where c = (,, ). () e) Check whether W {(x, x, x ) x+ x + x } m = is a subspace of n. () f) Consider the subspaces of : W = [{a, b}], W = [{b, c}], W = [{c, d}], where a = (,, ), b= (,, ), c= (,,), d= (,,). Check whether or not i) W = W, ii) W W = [{b}], iii) W + W =, iv) W W is a subspace of.

4 v). a) Let A Mn ( ) has at least two distinct elements. (7) W and X= { B M n ( ) BA= AB}. Prove that X is a real vector space of dimension greater than one. () b) Show that is an infinite dimensional vector space over. () V= f :[, ], V the set of even functions on [, ] and V the set of odd functions on [, ]. Prove that V ~ V V. (7) c) Let { } e e d) Define a linear transformation T : [x] : T(p(x)) = (p(), p( )). Check whether it is an isomorphism. () T ( x x+ x+ x, x x+ x x,x 5x+ x. a) Consider : : T(x, x, x, x ) = ). Obtain the matrix A, of T with respect to the bases and of and, respectively, where = {(7,,, ), (,,, ), (5,,,), (7,,, ) } and (,, ), (,, ),(,, ). = { } Also obtain the matrix B of T, with respect to the standard bases of and, and find the matrices P and Q such that A= PBQ. (6) b) Can the Gaussian elimination process be used for finding the solution of the following linear system? If so, apply it. If it can t be applied, use Cramer s rule for solving it. x + y+ z+ 6=, x+ y+ z+ =, x+ y+ 9z+ 6=. () c) Find the inverse of A = using elementary transformations. (5) 5. a) Prove that if λ is an eigenvalue of a non-singular matrix A, then A is an eigenvalue of λ Adj (A). () b) Give an example of a linear operator on whose minimal polynomial is ( x+ ). What could its characteristic polynomial be? () c) Check whether 5 is diagonalisable or not. ()

5 6 d) Using the Cayley-Hamilton theorem, check if A = 6 is invertible or not. If it is, find its inverse, If A does not exist, find Adj (A). (6) 6. a) Check whether the following functions define an inner product. i) <, > : : < (x, x), (y, y) > = xy x y. ii) <, > : : < (x, x ), (y, y ) > = xx yy. () b) Complete {(,, ), (,,)} to form a basis of c) Find process to obtain an orthonormal basis of. Then use the Gram-Schmidt from. (5) W, where is with respect to the standard inner product of, and {(x, x, x, x ) x+ x + 5x + x =, x+ x + x = } =. W Also obtain a non-trivial element of. () W i i i i 7. a) Find a unitary matrix whose first two columns are,,,,,,,. t t () b) Prove that the characteristic polynomial of a self-adjoint operator must have real coefficients. () 5