MATH 110 - Spring 2008 - Greicius - Midterm 1 (solution)

MATH 110 - Spring 2008 - Greicius - Midterm 1 (solution) -...

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Unformatted text preview: Math 110-S1 Spring 2008 Name: SID: Midterm 1 Write your name and SID on the front of your exam. You must JUSTIFY your answers, so show your work. Partial credit will be awarded even if answers are incorrect. No notes, books, or calculators. Good luck! 1. Define T : M 2 2 ( R ) M 2 2 ( R ) as T ( A ) = 1 1 1 1 A . a. (5 pts.) Prove T is linear and compute [ T ] , where is the standard basis of M 2 2 ( R ). b. (5 pts.) Give bases for N( T ) and R( T ). c. (5 pts.) Let = { 1 1- 1- 1 , 1- 1- 1 1 , 1 1 1 1 , 1- 1 1- 1 } . Compute [ T ] . d. (5 pts.) Find an invertible matrix Q such that [ T ] = Q- 1 [ T ] Q . You do NOT need to compute Q- 1 . SOLUTION: a. Linearity: Let C = 1 1 1 1 . Then T ( cA + B ) = C ( cA + B ) = cCA + CB = cT ( A )+ T ( B ). Next, a simple computation shows [ T ] = 1 0 1 0 0 1 0 1 1 0 1 0 0 1 0 1 ! . b. [ T ] row reduces to U = 1 1 0 1 0 1 0 0 0 0 ....
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MATH 110 - Spring 2008 - Greicius - Midterm 1 (solution) -...

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