MATH C192 - is invertible. Find a formula for T-1 and hence...

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BIRLA INSTITUTE OF TECHNOLOGY AND SCIENCE, PILANI I SEMESTER 2004 – 2005 COMPREHENSIVE EXAMINATION Course Number: MATH C192 Time: 3 Hours Course Title: MATHEMATICS II Dec.08, 2004 Max.Marks : 40 Note: There are TEN questions. All question carry EQUAL marks. 1. Let V be the Vector Space of polynomials a 0 + a 1 t + a 2 t 2 +…+ a n t n with real coefficients i.e. a i R .Determine whether or not W is a subspace of V where i. W consists of all polynomials with integral coefficients ii. W consists of all polynomials with degree 3 2. Let V be the Vector space of the function from R into R . Show that f, g, h V are linearly independent, where f (t) = e 2t , g (t) = t 2 , h (t) = 1. 3. Show that if F: V U and G: V U are linear, then F+G is linear. 4. Let T be a linear operator on R 3 defined by T (x, y, z) = (x-3y-2z, y-4z, z). Show that T
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Unformatted text preview: is invertible. Find a formula for T-1 and hence find T-1 (1,2,3) 5. Show that u (x, y) is harmonic in some domain and find a harmonic conjugate v (x, y) when u (x, y) = 2x x 3 + 3xy 2 6. If f (z) is defined by the equations f(z) = 1, y < 0 = 4y, y > 0 and C : z = -1 - i to z = 1 + i along the curve y = x 3 .Evaluate c dz z f ) ( 7. If f(z) = 1 / (z 2 + 2z + 2) and C: 1 = z ,Evaluate c dz z f ) ( 8. Represent the function f(z) = (z + 1) / (z 1) by its Laurent series for the domain 1 < z < 9. Evaluate =-+ 3 : 2 2 1 z C z z z dz, where C is a positively oriented simple closed contour. 10. Show that + 2 cos b a d = 2 2 2 b a-, a > b > 0...
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