Test2 - T ) = AntiSymm( R ) 4. Let S be an invertible n n...

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TEST 2 (Math 250 A) 1. (a) Show that the following map is a linear map: (20 pts) z y x z y x T R R T 4 3 2 ) , , ( : 3 (b) Is the following map linear? ) , ( ) , ( : 2 2 2 2 y x y x T R R T 2. (a) Explain why there is a unique linear map 2 2 : R R T for which T (1,-2) = (2,3) and T (3,1) = (0,1). (30 pts) (b) Find a formula for T above. (c) Is there a linear map 2 2 : R R T for which T (2,2) = (8,-6) and T (5,5) = (3,-2) ? ( Hint : Notice that (5,5) = 5/2(2,2) and explain why this causes a problem)
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3. Consider the linear map defined by: (20 pts) t A A A T R M R M T ) ( ) ( ) ( : 2 2 2 2 Show that: (a) Ker( T ) = Symm( R ) (b) Im(
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Unformatted text preview: T ) = AntiSymm( R ) 4. Let S be an invertible n n matrix and consider the linear map defined by: AS A T R M R M T n n n n ) ( ) ( ) ( : Show that T is one-to-one and onto. (20 pts) 5. Consider the linear map (differentiation) defined by: (10 pts) ) ( )) ( ( ] [ ] [ : 1 x p x p D x P x P D n n Find Ker( D ) and use that to conclude that D is onto. ( Hint : Use the Dimension Formula for the 2 nd half of the problem)...
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Test2 - T ) = AntiSymm( R ) 4. Let S be an invertible n n...

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