/*! 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 20 Let \(f, g, \in \mathcal{F}(R, R... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 20

Let \(f, g, \in \mathcal{F}(R, R)\) be the functions defined by \(f(t)=e^{r t}\) and \(g(t)=e^{s t}\), where \(r \neq s\). Prove that \(f\) and \(g\) are linearly independent in \(\mathcal{F}(R, R)\).

Short Answer

Expert verified
Assuming that the functions \( f(t) = e^{rt} \) and \( g(t) = e^{st} \) are linearly dependent, we find a contradiction in the non-zero constants c and d such that \( c\cdot e^{rt} + d\cdot e^{st} = 0 \). By choosing two different values for t, we show that this linear combination cannot be equal to zero for all t 鈭� 鈩�, proving that f(t) and g(t) are linearly independent in 鈩� 鈫� 鈩�.

Step by step solution

01

1. Define Linearly Dependent and Independent Functions:

For two functions f(t) and g(t) in 鈩� 鈫� 鈩�, they are linearly dependent if and only if there exist real numbers c and d not equal to 0 simultaneously such that: c鈭檉(t) + d鈭檊(t) = 0, for all t 鈭� 鈩�. If this condition is not satisfied, then the functions f(t) and g(t) are linearly independent.
02

2. Assume both functions are linearly dependent:

We will assume that f(t) and g(t) are linearly dependent, meaning there exist nonzero constants c and d such that the linear combination of the functions is equal to zero: c鈭檉(t) + d鈭檊(t) = 0 Substitute f(t) and g(t) as given in the problem: c鈭橽(e^{rt}\) + d鈭橽(e^{st}\) = 0
03

3. Analyze the Linear Combination:

Divide both sides of the equation by \( e^{r t} \): \( \frac{c\cdot e^{rt} + d\cdot e^{st}}{e^{rt}} \) = \( \frac{0}{e^{rt}} \) c + d鈭橽( e^{(s-r)t} \) = 0
04

4. Choose Appropriate Values for t:

We can pick two different values for t, say t鈧� and t鈧�, to create two separate equations: c + d鈭橽( e^{(s-r)t鈧亇 \) = 0 (Equation 1) c + d鈭橽( e^{(s-r)t鈧倉 \) = 0 (Equation 2) Since r 鈮� s, (s-r)鈮�0, and therefore, \(e^{(s-r)t鈧亇 \) 鈮� \(e^{(s-r)t鈧倉 \) when t鈧� 鈮� t鈧�.
05

5. Find Contradiction:

Subtract Equation 1 from Equation 2: d鈭�(\( e^{(s-r)t鈧倉 \) - \( e^{(s-r)t鈧亇 \)) = 0 Since \(e^{(s-r)t鈧亇 \) 鈮� \(e^{(s-r)t鈧倉 \), the term: (\( e^{(s-r)t鈧倉 \) - \( e^{(s-r)t鈧亇 \)) can't be equal to zero, which implies d = 0. This contradicts our assumption that both c and d are non-zero constants.
06

6. Conclusion:

Since we found a contradiction to our assumption that f(t) and g(t) are linearly dependent, it means that f(t) and g(t) are linearly independent in 鈩� 鈫� 鈩�.

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

Show that \(P_{n}(F)\) is generated by \(\left\\{1, x, \ldots, x^{n}\right\\}\).

Prove the following properties of the cross product: (a) \(u \times v=-(v \times u)\) (d) \(u \times(v+w)=(u \times v)+(u \times w)\) (b) \(u \times u=0\) for any vector \(u\) (e) \((v+w) \times u=(v \times u)+(w \times u)\) (c) \((k u) \times v=k(u \times v)=u \times(k v)\) \((\mathrm{f}) d(u \times v) \times w=(u \cdot w) v-(v \cdot w) u\)

In \(F^{n}\), let \(e_{j}\) denote the vector whose \(j\) th coordinate is 1 and whose other coordinates are 0 . Prove that $\left\\{e_{1}, e_{2}, \ldots, e_{n}\right\\}$ is linearly independent.

Let \(V\) denote the set of all differentiable real-valued functions defined on the real line. Prove that \(V\) is a vector space with the operations of addition and scalar multiplication defined in Example \(3 .\)

Let \(v_{1}, v_{2}, \ldots, v_{k}, v\) be vectors in a vector space \(\mathrm{V}\), and define \(\mathrm{W}_{1}=\) $\operatorname{span}\left(\left\\{v_{1}, v_{2}, \ldots, v_{k}\right\\}\right)$, and $\mathrm{W}_{2}=\operatorname{span}\left(\left\\{v_{1}, v_{2}, \ldots, v_{k}, v\right\\}\right)$. (a) Find necessary and sufficient conditions on \(v\) such that \(\operatorname{dim}\left(\mathrm{W}_{1}\right)=\) \(\operatorname{dim}\left(\mathrm{W}_{2}\right)\). (b) State and prove a relationship involving \(\operatorname{dim}\left(\mathrm{W}_{1}\right)\) and \(\operatorname{dim}\left(\mathrm{W}_{2}\right)\) in the case that $\operatorname{dim}\left(\mathrm{W}_{1}\right) \neq \operatorname{dim}\left(\mathrm{W}_{2}\right)$.
See all solutions

Recommended explanations on Math Textbooks

Logic and Functions

Read Explanation

Mechanics Maths

Read Explanation

Statistics

Read Explanation

Calculus

Read Explanation

Probability and Statistics

Read Explanation

Decision 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.