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Unformatted text preview: Solutions to First Midterm Exam MAT 310, Spring 2005 Name: ID#: Rec: problem 1 2 3 4 5 Total possible 20 20 20 20 20 100 score Directions: There are 5 problems on five pages in this exam. Make sure that you have them all. Do all of your work in this exam booklet, and cross out any work that the grader should ignore. You may use the backs of pages, but indicate what is where if you expect someone to look at it. Books, calculators, extra papers, and discussions with friends are not permitted. Feel free to confer with the psychic friends network if you can do so silently. However, I dont think Dionne Warwick knows much linear algebra. 1. (20 points) Let V be the (real) vector space of all functions f from R into R . a.) Is W = ' f  f ( 2 ) = f (2) a subspace of V ? Prove or give a reason why not. Solution: Yes, it is a subspace. To prove this, let f and g be functions in W and let c be any scalar. We need to see that the function cf + g is also in W . But ( cf + g )( 2 ) = cf ( 2 ) + g ( 2 ) = cf (2) + g (2) = ( cf + g )(2) b.) Is W = ' f  [ f ( )] 2 = f (2) a subspace of V ? Prove or give a reason why not. Solution: No, it is not, since [( cf )( )] 2 = c 2 [ f ( )] 2 = c 2 f (2) 6 = cf (2) unless c = 0 or c = 1 (or f (2) = 0 ). 2. (20 points) Let T be the transformation from C 3 to C 3 corresponding to the matrix 1 0 i 0 1 1 i 1 0 a) What is the rank of T ? Write a basis for the image of T ....
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