Math121PS1

Math121PS1 - Math 121 Homework, Week 1 Michael Von Kor...

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Math 121 Homework, Week 1 Michael Von Korﬀ October 1, 2004 Problem 1. A real-valued function f deﬁned on the real line is called an even function if f ( - x ) = f ( x ) for each real number x . Prove that the set of even functions deﬁned on the real line with the operations of addition and scalar multiplication deﬁned in Example 3 is a vector space. Proof. We could show each of the 8 vector space requirements as well as closure under addition and scalar multiplication; together, all of these would show that the set is a vector space. But there’s an easier way: Let S be the set of even, real-valued functions. Then S is a subset of the set of all real-valued functions, which is a vector space by Example 3. Thus, in order to show that S is a vector space, we need only show that it is a subspace of the set of all real-valued functions. To do this, we need only apply Theorem 1.3 and show that S is closed under addition and scalar multiplication. That is, we must show that f,g S, λ R ,f + g S and λf S . But x R , ( f + g )( x ) = f ( x ) + g ( x ) = f ( - x ) + g ( - x ) = ( f + g )( - x ) so f + g is even, and S is closed under addition. Similarly, ( λf )( x ) = λ ( f ( x )) = λ ( f ( - x )) = ( λf )( - x ) so λf is even, and S is closed under scalar multiplication. Thus S is a vector subspace of the set of real-valued functions. Problem 2. Let V denote the set of all m × n matrices with real number entries. V is a vector space over the ﬁeld R of real numbers under the usual deﬁnitions of matrix addition and multiplication. Is V a vector space over the ﬁeld Q of rational numbers under the same addition and multiplication?

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This note was uploaded on 02/22/2012 for the course EE 441 taught by Professor Neely during the Spring '08 term at USC.

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Math121PS1 - Math 121 Homework, Week 1 Michael Von Kor...

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