solution5 - pp 73 PExercise 1 Homework assignment 5 Which...

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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 ) ; No, because T (0 , 0) 6 = (0 , 0) . (b) T ( x 1 , x 2 ) = ( x 2 , x 1 ); Yes, because x 2 and x 1 are linear homogeneous functions of x 1 , x 2 . (c) T ( x 1 , x 2 ) = ( x 2 1 , x 2 ) ; No, because, say, 2 T (1 , 0) 6 = T (2 , 0) . (d) T ( x 1 , x 2 ) = (sin x 1 , x 2 ); No, since 2 T ( π 2 , 0) = (2 , 0) 6 = (0 , 0) = T ( π, 0) . (e) T ( x 1 , x 2 ) = ( x 1 - x 2 , 0) . Yes, because x 1 - x 2 and 0 are linear homogeneous functions of x 1 , x 2 . Find the range, rank, null space, and nullity for th e differen- tiation 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. The range of D consists of all polynomials of degree strictly less than k , since any polynomial p ( x ) = a n x n + . . . + a 0 is the derivative of the polynomial R p ( x ) = a n n +1 x n +1 + . . . + a 0 x . The null space of D consists of all constants. Hence, the rank of D is k , and the nullity is 1 . The range of T consists of all continuous functions f such that f has con- tinuous first derivative and f (0) = 0 . The null space of T is trivial, because if a function is not identically zero then so is its integral. Hence, the rank of T is infinite, and the nullity is 0 . Let F be a subfield of the complex numbers and let T be the function from F 3 into F 3 defined by T ( x 1 , x 2 , x 3 ) = ( x 1 - x 2 + 2 x 3 , 2 x 1 + x 2 , - x 1 - 2 x 2 + 2 x 3 ) (a) Verify that T is a linear transformation. (b) If ( a, b, c ) is a vector in F 3 , what are the conditions on a , b and c that the vector be in the range of T ? What is the rank of T ? (c) What are the conditions on a , b , and c that ( a, b, c ) be in the null space of T ? What is the nullity of T ?
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