This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
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...
View
Full
Document
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.
 Spring '08
 Davidson,M
 Math

Click to edit the document details