MATH 2107assign1

MATH 2107assign1 - MATH 2107 LINEAR ALGEBRA II ASSIGNMENT 1...

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MATH 2107 LINEAR ALGEBRA II ASSIGNMENT 1 DUE: October 16 at the beginning of the tutorial 1. Let 3 2 : P P T be a linear transformation s.t. 3 2 ) ( x x T = , 0 ) 1 ( = + x T , x x T = ) 1 (. Find T and determine ) 1 ( 2 + + x x T . 2. Show that the function ) , ( ) , ( xy y x y x T + = is not a linear transformation. 3. Let 3 4 : T be a function given by ) 7 5 4 2 , 5 4 2 , 4 3 2 ( ) , , , ( w z y x w z y x w z y x w z y x T + + + + + + + + = a) Show that T is a linear transformation. b) Find ) ker( T and a basis for ) ker( T . c) Determine the nullity of T and rank of T . 4. Determine which of the following linear transformation is (i) one-to-one (ii) onto (iii) isomorphism a) 3 3 : T given by } , 3 , { ) , , ( z y y z z y x T = b) 3 3 : T given by } , , { ) , , ( x z z y y x z y x T + + + = c) n P T : given by n n n r x r x r x r r T = + + + + ) ( 2 2 1 0 K d) 3 2 : T given by } , , { ) , ( y x y x y x T + = 5. Given } , , { 2 1 n v v v K in a vector space V , let V T n : given by n n n v r v r r r T , ) , , ( 1 1 1 K K + = . a) Show that
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