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**Unformatted text preview: **MATH 601, AUTUMN 2008 Additional homework III, October 18 PROBLEMS 1. Determine dim T ( R 3 ) for the linear operator T : R 3 R 4 given by T x y z = 1- 1 1 2 6- 1 3 3 1 1 5 x y z , with vectors of R 3 and R 4 written in the column form. 2. For any integer n 3, find dim W , where W denotes the vector space of all polynomials f of the real variable x with deg f n and f (1) = f (0) = R 1 f ( x ) dx = 0. 3. Given an integer n 0, let W be the vector space of all polynomials f of the complex variable z with deg f n and f (0) = 0. Find det T , where T : W W is the linear operator given by ( Tf )( z ) = zf ( z ). 4. Let a unit of measure | | in a 4-dimensional real vector space V be chosen in such a way that | K | = 7, where K is the parallelepiped in V with the given edge vectors p, q,r,s V emanating from 0. Evaluate | K | for the parallelepiped K in V with the edge vectors u, v,w,z such that u = p- s , v = p + q + r , w = p- r and z = q + s . 5. Denoting | | the standard unit of measure of R 3 , find the volume | K | , where K is the image of the cube { ( x,y, z ) R 3 : x,y, z [0 , 1] } under the linear operator T : R 3 R 3 defined by T x y z = 2- 1 3 1 2 2 2 x y z , with vectors of R 3 written in the column form. 6. In the 3-dimensional real vector space V of all C 3 functions f : R R with f 000 = 0 everywhere, define the volume form by ( f,g,h ) = f (1) g (2) h (3)- f (1) g (3) h (2) + f (2) g (3) h (1)- f (2) g (1) h (3) + f (3)...

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