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finalexamsol - Math 115a Final Exam Lecture 3 Winter 2009...

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Unformatted text preview: Math 115a Final Exam Lecture 3 Winter 2009 Name: Instructions: • There are 8 problems. Make sure you are not missing any pages. • Unless stated otherwise, you may use without proof anything proven in the sections of the book covered by this test (excluding the exercises). • Give complete, convincing, and clear answers (or points will be deducted). • No calculators, books, or notes are allowed. • Answer the questions in the spaces provided on the question sheets. If you run out of room for an answer, continue on the back of the page. Question Points Score 1 10 2 10 3 10 4 10 5 10 6 10 7 10 8 10 Total: 80 1. (10 points) Let V be a vector space, let T : V → V be the zero transformation (that is, T ( v ) = 0 v for every v ∈ V ), and let T : V → V be any linear transformation. Prove that T 2 = T ⇔ range( T ) ⊂ nullspace( T ) Solution: ⇒ Suppose T 2 = T and y ∈ range( T ); we need to show that y ∈ nullspace( T ) . By definition of range, there is an x ∈ V such that T ( x ) = y. Since T 2 = T we have 0 V = T 2 ( x ) = T ( T ( x )) = T ( y ) , and so y ∈ nullspace( T ) . ⇐ Suppose that range( T ) ⊂ nullspace( T ) and that x ∈ V . We need to show that T 2 ( x ) = V . Since T ( x ) ∈ range( T ), we have T ( x ) ∈ nullspace( T ) and so T 2 ( x ) = T ( T ( x )) = 0 V . 2. (10 points) Let V be a vector space, and let T : V → V and U : V → V be linear...
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This note was uploaded on 01/27/2010 for the course MATH 7330 taught by Professor Davidson,m during the Spring '08 term at LSU.

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finalexamsol - Math 115a Final Exam Lecture 3 Winter 2009...

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