BACHELOR'S DEGREE PROGRAMME (BDP) Term-End Examination. June, 2018 ELECTIVE COURSE : MATHEMATICS MTE-02 : LINEAR ALGEBRA

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1 No. of Printed Pages : 8 1MTE-021 BACHELOR'S DEGREE PROGRAMME (BDP) Term-End Examination 046OB June, 2018 ELECTIVE COURSE : MATHEMATICS MTE-02 : LINEAR ALGEBRA Time : 2 hours Maximum Marks : 50 (Weightage : 70%) Note : Attempt five questions in all. Question no. 7 is compulsory. Answer any four questions from questions no. 1 to 6. Use of calculators is not allowed. 1. (a) Check that the set W = {(x1, x2, x3, x4) I x1 + x2 = 2x 3 } is a subspace of R4. Find a, basis of W. Hence, find the dimension of W. 5 (b) Let T : R3 R3 be the linear transformation defined by T(x, y, z) = (2x + y z, x + y, x + z). Find the matrix of the transformation with respect to the ordered basis {v 1, v2, v3}, where v 1 = (1, 1, 1), v 2 = (1, 1, 0) and v3 = (1, 0, 0). Is T invertible or not? Justify your answer. 5 MTE-02 1 P.T.O.

2 2. (a) Let T : R4 --> R4 be the linear transfoi-mation defined by T(x1, X2, X3, X4) = ( x2, X1, x4, X3). Check that T4 = I. Also, find the minimal polynomial of T. 5 (b) Find the inverse of the matrix A= using row reduction (a) Check whether the following system of equations can be solved using Cramer's rule : 2x + 3y + z = 11 x + y + 2z = 6 2x y + 2z = 4 If the system can be solved by the rule, then use it to obtain the solution. If the system cannot be solved using Cramer's rule, use Gaussian elimination to solve it. 5 (b) Find the radius and the centre of the circular section of the sphere r = 4, cut off by the plane r. (2i j + 4k) = 3. 5 MTE-02 2

3 4. (a) Check whether or not the matrix A= is diagonalisable. 5 (b) Check that T : R 3 -> R3, defined by T(x1, x2, x3) = (x1 + x3, x2 + 2x3, X1 - X2 - x3) is a linear operator. Also find the null space of T. 5. (a) Check whether the function f : R -> R, defined by f(x) = x3 + 1, is 1-1. Is it onto? Justify your answers. (b) Consider the linear operator T : C4 -> C4, defined by T(z 1, z2, z3, z4) = (- i22, iz1, - iz4, z3). Find T*(w 1, w2, w3, w4), where wi E C Vi= 1, 2, 3, 4 and check whether T is self-adjoint under the standard inner product on C4. Further, check whether T is unitary (a) Find the orthogonal canonical reduction of the quadratic form - x2 + y2 + z2 + 2xy - 2xz + 2yz. Also, find its principal axes. 7 (b) Check whether or not i + j - 2k j± k and form an 3 at 2 orthonormal basis for R3. 3 MTE-02 3 P.T.O.

4 7. Which of the following statements are True and which are False? Justify your answers with a short proof or by a counter-example. 5x2=10 (a) (b) (c) The relation defined in R by 'x y if x y' is an equivalence relation. There is no system of linear equations over R that has exactly two solutions. If the characteristic polynomials of two matrices are equal, their minimal polynomials are also equal. (d) The determinant of any unitary matrix is 1. (e) Any two real quadratic forms of the same rank are equivalent over R. MTE-02 4

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