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Math254Mid3 - Math 254 Name MIDTERM 3 MATH 254 SUMMER 2002...

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Math 254 Name: ________________________ MIDTERM 3 MATH 254 - SUMMER 2002 - KUNIYUKI CHAPTERS 6, 7 GRADED OUT OF 75 POINTS ¥ 2 = 150 POINTS TOTAL Circle your final answers! Show all work and simplify wherever appropriate, as we have done in class! A scientific calculator is allowed on this exam. 1) The linear transformation T R R : 2 3 Æ is such that T 1 1 3 4 1 , , , ( ) ( ) = - and T 0 1 1 5 3 , , , ( ) ( ) = . Find T 4 5 , ( ) . Hint: Remember the definition of a linear transformation. (5 points) 2) The linear transformation T R R : 2 2 Æ is such that T v v v v v v 1 2 1 2 1 2 2 2 3 , , ( ) ( ) = - + . Find the preimage of 17 5 , ( ) . (7 points)
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3) The linear transformation T R R : 5 7 Æ is such that dim Ker T ( ) ( ) = 3 . (11 points total) a) What is the domain of T ? b) What is nullity T ( ) ? c) What is rank T ( ) ? d) True or False: Range T ( ) is a subspace of R 7 . Circle one: True False e) True or False: Ker T ( ) is a subspace of R 7 . Circle one: True False 4) T R R : 2 2 Æ is a linear transformation such that, relative to the standard basis of R 2 , T x y T x y x y , , ( ) ( ) = - + 3 4 . Another basis for R 2 is given by: ¢ = ( ) ( ) { } B 1 3 2 0 , , , . (13 points total) a)
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