Solution Found!
(Calculus required) Define as follows:For be the
Chapter 4, Problem 34E(choose chapter or problem)
Problem 34E
(Calculus required) Define as follows:
For be the antiderivative F of f such that F(0) = 0. Show that T is a linear transformation, and describe the kernel of T. (See the notation in Exercise 20 of Section 4.1.)
Reference:
The set of all continuous real-valued functions defined on a closed interval [a, b] in is denoted by C.[a, b]. This set is a subspace of the vector space of all real-valued functions defined on [a, b].
a. What facts about continuous functions should be proved in order to demonstrate that C.[a, b] is indeed a subspace as claimed? (These facts are usually discussed in a calculus class.)
b. Show that is a subspace of C.[a, b]
Questions & Answers
QUESTION:
Problem 34E
(Calculus required) Define as follows:
For be the antiderivative F of f such that F(0) = 0. Show that T is a linear transformation, and describe the kernel of T. (See the notation in Exercise 20 of Section 4.1.)
Reference:
The set of all continuous real-valued functions defined on a closed interval [a, b] in is denoted by C.[a, b]. This set is a subspace of the vector space of all real-valued functions defined on [a, b].
a. What facts about continuous functions should be proved in order to demonstrate that C.[a, b] is indeed a subspace as claimed? (These facts are usually discussed in a calculus class.)
b. Show that is a subspace of C.[a, b]
ANSWER:
Solution 34E
Step 1 of 3
Let the mapping be defined as is an antiderivative F of f such that, where.
The objective is to show that T is a linear transformation.
Let f and g be any elements in and let c be any scalar.
Then is the antiderivative F of f with and is the antiderivative G of g with.
By the rules of antidifferentiation is an antiderivative of
Thus
