MATH 2107assign1sol

MATH 2107assign1sol - MATH 2107 LINEAR ALGEBRA II...

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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 . The set } 1 , 1 , { 2 + x x x is a basis of 2 P , so every vector 2 cx bx a p + + = in 2 P is a linear combination of these vectors. i.e. ) 1 ( ) 1 ( ) ( 1 1 2 1 2 + + + = + + = x c x b x a cx bx a p for some scalar 1 a , 1 b , 1 c . i.e. 2 1 1 1 1 1 2 ) ( ) ( x a x c b c b cx bx a + + + = + + i.e. we need to solve + 2 / ) ( 2 / ) ( 1 0 0 0 1 0 0 0 1 ~ 0 0 1 1 1 0 1 1 0 a b b a c c b a ( several steps of row operations need to be taken to reach the final RREF of the matrix). i.e. ) 1 ( 2 ) ( ) 1 ( 2 ) ( ) ( 2 2 + + + + = + + = x a b x b a x c cx bx a p So, ) 1 ( 2 ) ( ) 1 ( 2 ) ( ) ( ) ( 2 + + + + = x T a b x T b a x cT p T x a b cx x a b b a cx cx bx a T 2 ) ( 2 ) ( 0 2 ) ( ) ( 3 3 2 + = + + + = + + So, 3 3 2 2 ) 1 1 ( . 1 ) 1 ( x x x x x T = + = + + 2. Show that the function ) , ( ) , ( xy y x y x T + = is not a linear transformation. I. Let ) , ( 1 1 1 y x v = and ) , ( 2 2 2 y x v = So, ) , ( ) ( 1 1 1 1 1 y x y x v T + = , ) , ( ) ( 2 2 2 2 2 y x y x v T + = ) , ( 2 1 2 1 2 1 y y x x v v + + = + And )) )( ( , ( ) ( 2 1 2 1 2 1 2 1 2 1 y y x x y y x x v v T + + + + + = + ) , ( ) ( ) ( 2 2 1 1 2 1 2 1 2 1 y x y x y y x x v T v T + + + + = + And, ) ( ) ( ) ( 2 1 2 1 v T v T v v T + + . So the given function is not closed under vector addition. II. Let ) , ( y x v = . So ) , ( ay ax av = ) , ( ) ( 2 xy a ay ax av T + = ) , ( ) ( xy y x v T + = ) , ( ) ( axy ay ax v aT + = So the given function is not closed under scalar multiplication. Hence the given function is not a linear transformation. (By showing that any one of the operations is not satisfied, you can claim that it is not a linear transformation)
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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. Let ) , , ( 1 1 , 1 1 1 w z y x v = and ) , , ( 2 2 , 2 2 2 w z y x v = be two vectors in 4 , I. Then ) , , , ( 2 1 2 1 2 1 2 1 2 1 w w z z y y x x v v + + + + = + ), ( 4 ) ( 3 ) ( 2 ) (( ) ( 2 1 2 1 2 1 2 1 2 1 w w z z y y x x v v T + + + + + + + = + ), ( 5 ) ( 4 ) ( 2 ) ( 2 1 2 1 2 1 2 1 w w z z y y x x + + + + + + )) ( 7 ) ( 5 ) ( 4 ) ( 2 2 1 2 1 2 1 2 1 w w z z y y x x + + + + + + + ) 7 5 4 2 , 5 4 2 , 4 3 2 ( ) ( 1 1 1 1 1 1 1 1 1 1 1 1 1 w z y x w z y x w z y x v T + + + + + + + + = ) 7 5 4 2 , 5 4 2 , 4 3 2 ( ) ( 2 2 2 2 2 2 2
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MATH 2107assign1sol - MATH 2107 LINEAR ALGEBRA II...

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