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

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

{[ snackBarMessage ]}

Page1 / 2

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

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online