# solution8 - Homework assignment 8 Section 3.7 pp 115...

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Unformatted text preview: Homework assignment 8 Section 3.7 pp. 115 Exercise 2. Let V be the vector space of all polynomial functions over the field of real numbers. Let a and b be fixed real numbers and let f be the linear functional on V defined by f ( x ) = Z b a p ( x ) dx If D is the differentiation operator on V , what is D t f ? Solution: By the fundamental theorem of calculus, we get: D t f ( q ) = f ( D ( q )) = Z b a q ( x ) dx = q ( b )- q ( a ) Exercise 3. Let V be the space of all n × n matrices over a field F and let B be a fixed n × n matrix. If T is the linear operator on V defined by T ( A ) = AB- BA and if f is the trace function, what is T t f ? Solution: Using the result from the previous hw. tr( AB ) = tr( BA ) we get: T t f ( A ) = f ( T ( A )) = tr( T ( A )) = tr( AB- BA ) = tr( AB )- tr( BA ) = 0 Exercise 6. Let n be a positive integer and let V be the space of all polyno- mial functions over the field of real numbers which have degree at most n , i.e., functions of the form f ( x ) = c + ... + c...
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solution8 - Homework assignment 8 Section 3.7 pp 115...

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