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Pune University - SY BSc (Sem - II) MATHEMATICS - I, April 2010

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Total No. of Questions : 4] P479 [Total No. of Pages : 2 [3717] - 201 S.Y. B.Sc. (Sem. - II) MATHEMATICS Linear Algebra (New Course) (Paper - I) Time : 2 Hours] [Max. Marks : 40 Instructions to the candidates : 1) All questions are compulsory. 2) Figures to the right indicate full marks. Q1) Solve the following questions. [10] a) Is the set {(1, 1), (2, 1), (3, 0)} linearly dependent? Justify. b) Determine whether W = {(x, y, z): x + y + z = 1} is a subspace of R3. c) If T : R2 R3 is a linear transformation such that T(1, 0) = (2, 1, 1) and T (0, 1) = (1, 1, 0), then find T(2, 1). d) If u and v are orthogonal vectors in an inner product space, then prove that u + v 2 = u 2 + v 2 . e) Find the orthogonal projection of v along u where v = (1, 1, 0) and u = (0, 1, 1). f) 0 Find the eigen values of the matrix 1 1 . 0 g) If T : R3 R2 is given by T(x, y, z) = (x y, y z), then find the nullity of T. h) Compute the angle between the vectors (1, 0) and (1, 1). i) If S = {(x, y, z) : 2x y + z = 0}, then find the dimension of S. j) Construct an orthonormal basis of R3 which is not a standard basis. Q2) Attempt any two of the following: [10] a) Show that any orthonormal set in an inner product space is linearly independent. b) Let T : R2 R3 be defined by T(x, y) = (x, x + y, y). Find the range and kernel of T. c) Find a basis of R3 which contains the vector (1, 1, 1). P.T.O. Q3) Attempt any two of the following: a) [10] If T : R3 R3 is defined by T(x, y, z) = (x + z, x + y + 2z, 2x + y + 3z), then find dim ker(T) and use Rank - Nullity theorem to find dim Im(T). b) State and prove Cauchy - Schwarz inequality. c) Find the eigen values and eigen vectors of the matrix 0 0 2 0 2 2 0 . 0 3 Q4) Attempt any one of the following: a) [10] Apply Gram - Schmidt process to the set {( 1, 0, 1), (1, 1, 0), (0, 0, 1)} to obtain an orthonormal basis of R3. ii) If {e1, . . . . ., en} is a basis of a vector space V, then show that {e1, . . . , en} is a maximal linearly independent set. i) If V and W are finite dimensional vector spaces and T : V W is a linear transformation, then prove that dim V = dim ker (T) + dim Im (T). ii) b) i) Show that the reflection matrix. cos2 R = sin2 sin2 cos2 is diagonalizable. Y [3717] - 201 2

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Additional Info : S.Y. B.Sc. Mathematics (Semister - I) : Linear Algebra (Paper - I) (New Course), Pune University
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