This preview shows pages 1–3. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: Math 115a Midterm 1 Lecture 1 Fall 2009 Name: Signature: Instructions: There are 5 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. You may only cite an exercise from the book if it was assigned as homework. Question Points Score 1 10 2 10 3 10 4 15 5 15 Total: 60 1. (10 points) Recall that P 5 ( R ) is the vector space of polynomials of degree less than or equal to 5. Let A = { f P 5 ( R ): f (0) = 0 } and B = { f P 5 ( R ): f (1) = 0 } . Prove that A B is a subspace of P 5 ( R ) . Solution: Let p ( x ) = 0 for all x R . Certainly p (0) = 0, so p A . Also, p (1) = 0, so p B . Hence p A B . This proves that the zero polynomial is in A B . Now let f,g A B and R . We know ( f + g )( x ) = f ( x ) + g ( x ) for any x R , by the definition of addition in P 5 ( R ). In particular, ( f + g )(0) = f (0) + g (0) = 0 + 0 = 0 so f + g A , and ( f + g )(1) = f (1) + g (1) = 0 + 0 = 0 so f + g B . Hence f + g A B . Similarly, ( f )( x ) =...
View
Full
Document
This note was uploaded on 02/22/2010 for the course MATH Math 115 taught by Professor Taft during the Fall '06 term at UCLA.
 Fall '06
 Taft
 Math, Linear Algebra, Algebra

Click to edit the document details