hw7 - space over the eld of real numbers (with usual...

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Homework assignment 7 Due date: October 20 pp.73-74 Exercise 1. Which of the following maps T from R 2 into R 2 are linear transformations? (a) T ( x 1 , x 2 ) = (1 + x 1 , x 2 ); (b) T ( x 1 , x 2 ) = ( x 2 , x 1 ); (c) T ( x 1 , x 2 ) = ( x 2 1 , x 2 ); (d) T ( x 1 , x 2 ) = (sin x 1 , x 2 ); (e) T ( x 1 , x 2 ) = ( x 1 - x 2 , 0). Exercise 3. Find the range, rank, null space, and nullity for the di±er- entiation transformation D on the space of polynomials of degree k : D ( f ) = f 0 . Do the same for the integration transformation T : T ( f ) = Z x 0 f ( t ) dt. Exercise 5. If α 1 = (1 , - 1) , β 1 = (1 , 0) α 2 = (2 , - 1) , β 2 = (0 , 1) α 3 = ( - 3 , 2) , β 3 = (1 , 1) is there a linear transformation T from R 2 into R 2 such that T α i = β i for i = 1 , 2 , 3? Exercise 6. Describe in coordinates (as in Exercise 1) the linear trans- formation T from F 2 into F 2 such that T e 1 = ( a, b ) , T e 2 = ( c, d ) ( { e 1 , e 2 } is the standard basis in F 2 ). Exercise 8. Describe explicitly the linear transformation from R 3 into R 3 that has as its range the subspace spanned by (1 , 0 , - 1) and (1 , 2 , 2). Exercise 10. Let V be the set of comlex numbers regarded as a vector
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Unformatted text preview: space over the eld of real numbers (with usual operations). Find a function from V into V that is a linear transformation on V , but that is not a linear transformation on C 1 , i.e., that is not complex linear. Exercise 13. Let V be a vector space and T a linear transformation from V into V . Prove that the following two statements about V are equivalent. (a) The intersection of the range of T and the null space of T is the zero subspace of V . 1 (b) If T ( T ) = 0, then T = 0. Bonus exercise 12. Let V be an n-dimensional vector space over the Feld F and let T be a linear transformation from V into V such that the range and null spaces of T are identical. Prove that T is even. (Can you give an example of such a linear transformation T ?) 2...
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hw7 - space over the eld of real numbers (with usual...

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