/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 10 For each \(f \in C[0,1]\), defin... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 4: Problem 10

For each \(f \in C[0,1]\), define \(L(f)=F,\) where \\[ F(x)=\int_{0}^{x} f(t) d t \quad 0 \leq x \leq 1 \\] Show that \(L\) is a linear operator on \(C[0,1]\) and then find \(L\left(e^{x}\right)\) and \(L\left(x^{2}\right)\)

Short Answer

Expert verified
\(L\) is a linear operator as it satisfies both conditions \(L(f + g) = L(f) + L(g)\) and \(L(cf) = cL(f)\). We find \(L(e^x) = e^x - 1\) and \(L(x^2) = \frac{1}{3}x^3\).

Step by step solution

01

Show that L is a linear operator on the set of continuous functions C[0, 1]

Recall that an operator \(L\) is linear if it satisfies the following conditions for any functions \(f, g \in C[0, 1]\) and any scalar \(c \in \mathbb{R}\): 1. \(L(f + g) = L(f) + L(g)\) 2. \(L(cf) = cL(f)\) To show that \(L\) is a linear operator, we will verify these two conditions: 1. \(L(f + g) = \int_{0}^{x}(f(t) + g(t)) dt = \int_{0}^{x} f(t) dt + \int_{0}^{x} g(t) dt = L(f) + L(g)\) 2. \(L(cf) = \int_{0}^{x} cf(t) dt = c \int_{0}^{x} f(t) dt = cL(f)\) Both conditions are satisfied, so \(L\) is indeed a linear operator on \(C[0, 1]\).
02

Compute L for the function e^x

Now we must compute the operator \(L\) for \(e^x\). Using the definition of \(L\), we have: \(L(e^{x}) = F(x) = \int_{0}^{x} e^t dt\) This integral is straightforward to compute, as the antiderivative of \(e^t\) is simply \(e^t\). Thus, integrating within the limits of the integral, we have: \(L(e^{x}) = e^x |_{0}^{x} = e^x - e^0 = e^x - 1\)
03

Compute L for the function x^2

Finally, we must compute the operator \(L\) for the function \(x^2\). Using the definition of \(L\), we have: \(L(x^2) = F(x) = \int_{0}^{x} t^2 dt\) The antiderivative of \(t^2\) is \(\frac{1}{3}t^3\), so we can compute the integral using the limits of the integral: \(L(x^2) = \frac{1}{3}x^3 |_{0}^{x} = \frac{1}{3}x^3 - \frac{1}{3}(0)^3 = \frac{1}{3}x^3\) So, we have successfully found that \(L(e^x) = e^x - 1\) and \(L(x^2) = \frac{1}{3}x^3\).

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Let \(L_{1}: U \rightarrow V\) and \(L_{2}: V \rightarrow W\) be linear transformations, and let \(L=L_{2} \circ L_{1}\) be the mapping defined by \\[ L(\mathbf{u})=L_{2}\left(L_{1}(\mathbf{u})\right) \\] for each \(\mathbf{u} \in U .\) Show that \(L\) is a linear transformation mapping \(U\) into \(W\)

For each of the following linear operators \(L\) on \(\mathbb{R}^{3}\), find a matrix \(A\) such that \(L(\mathbf{x})=A \mathbf{x}\) for every \(\mathbf{x}\) in \(\mathbb{R}^{3}:\) (a) \(L\left(\left(x_{1}, x_{2}, x_{3}\right)^{T}\right)=\left(x_{3}, x_{2}, x_{1}\right)^{T}\) (b) \(L\left(\left(x_{1}, x_{2}, x_{3}\right)^{T}\right)=\left(x_{1}, x_{1}+x_{2}, x_{1}+x_{2}+x_{3}\right)^{T}\) (c) \(L\left(\left(x_{1}, x_{2}, x_{3}\right)^{T}\right)=\left(2 x_{3}, x_{2}+3 x_{1}, 2 x_{1}-x_{3}\right)^{T}\)

Let a be a fixed nonzero vector in \(\mathbb{R}^{2} .\) A mapping of the form \\[ L(\mathbf{x})=\mathbf{x}+\mathbf{a} \\] is called a translation. Show that a translation is not a linear operator. Illustrate geometrically the effect of a translation.

Let \(E=\left\\{\mathbf{u}_{1}, \mathbf{u}_{2}, \mathbf{u}_{3}\right\\}\) and \(F=\left\\{\mathbf{b}_{1}, \mathbf{b}_{2}\right\\},\) where \\[ \mathbf{u}_{1}=\left(\begin{array}{r} 1 \\ 0 \\ -1 \end{array}\right), \quad \mathbf{u}_{2}=\left(\begin{array}{l} 1 \\ 2 \\ 1 \end{array}\right), \quad \mathbf{u}_{3}=\left(\begin{array}{r} -1 \\ 1 \\ 1 \end{array}\right) \\] and \\[ \mathbf{b}_{1}=(1,-1)^{T}, \quad \mathbf{b}_{2}=(2,-1)^{T} \\] For each of the following linear transformations \(L\) from \(\mathbb{R}^{3}\) into \(\mathbb{R}^{2},\) find the matrix representing \(L\) with respect to the ordered bases \(E\) and \(F\) (a) \(L(\mathbf{x})=\left(x_{3}, x_{1}\right)^{T}\) (b) \(L(\mathbf{x})=\left(x_{1}+x_{2}, x_{1}-x_{3}\right)^{T}\) (c) \(L(\mathbf{x})=\left(2 x_{2},-x_{1}\right)^{T}\)

Show that each of the following are linear operators on \(\mathbb{R}^{2}\). Describe geometrically what each linear transformation accomplishes. (a) \(L(\mathbf{x})=\left(-x_{1}, x_{2}\right)^{T}\) (b) \(L(\mathbf{x})=-\mathbf{x}\) (c) \(L(\mathbf{x})=\left(x_{2}, x_{1}\right)^{T}\) (d) \(L(\mathbf{x})=\frac{1}{2} \mathbf{x}\) (e) \(L(\mathbf{x})=x_{2} \mathbf{e}_{2}\)
See all solutions

Recommended explanations on Math Textbooks

Discrete Mathematics

Read Explanation

Statistics

Read Explanation

Probability and Statistics

Read Explanation

Decision Maths

Read Explanation

Calculus

Read Explanation

Mechanics Maths

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.