Midterm1_practice_sols - Solutions to Practice Midterm 1 By...

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Solutions to Practice Midterm 1 By Chu-Wee Lim Author’s note: I’ve attempted to be as complete as possible in my solution, hence the wordiness. In the midterm, you can probably miss one or two small steps. (Question 1): True or False (with justifcation): I± A and B are n -by- n matrices with entries ±rom F ,then AB =0i±andonly BA =0 . Solution. False. Counterexample: i± A =( 00 01 )and B =( 01 00 ), then AB =( 00 00 ) but BA = ( 01 00 ). (Question 2): True or False (with justifcation): I± A and B are n -by- n matrices with entries ±rom R ,then AB =7 I n i± and only BA =7 I n . Solution. True. First, a general statement. We claim that a linear map T : R n R n is an isomorphism i² it is one-to-one. Now, is obvious. For the converse, suppose T is one-to-one. Then n =d im ( R n )=d im N ( T )+dim R ( T ). Since T is one-to-one, dim N ( T ) = 0, and so dim R ( T )= n .S inc e R ( T ) R n , this implies R ( T )= R n . Now look at L A and L B as linear maps R n R n . By symmetry, it suffices to prove = . Since L A L B =7 · 1 V is one-to-one, L A is also one-to-one. By the above paragraph, L A is an isomorphism and thus has an inverse U , i.e. UL A = L A U =1 V .So , UL A L B = L B = 7 U = L B = U = 1 7 L B . Thus, 1 7 L B is the inverse o± L A and 1 7 L B L A =1 V = L B L A =7 · 1 V . (Question 3): True or False (with justifcation): I± x, y V and a, b F ,then ax + by =0i± and only i± x is a scalar multiple o±
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This note was uploaded on 07/19/2008 for the course MATH 110 taught by Professor Gurevitch during the Summer '08 term at University of California, Berkeley.

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Midterm1_practice_sols - Solutions to Practice Midterm 1 By...

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