AmoahMaxy
Answered

Discover the answers to your questions at Westonci.ca, where experts share their knowledge and insights with you. Get quick and reliable solutions to your questions from a community of experienced professionals on our platform. Discover in-depth answers to your questions from a wide network of professionals on our user-friendly Q&A platform.

Exercise 8.7

Suppose [tex]L[/tex] is a linear operator, that is, [tex]L(\alpha u + \beta v) = \alpha L(u) + \beta L(v)[/tex]. Consider the homogeneous and nonhomogeneous linear equations

[tex]\[
\begin{array}{l}
L u = 0 \\
L u = f
\end{array}
\][/tex]

where [tex]f[/tex] is some function. Suppose [tex]v[/tex] is a solution to the homogeneous equation, and [tex]w[/tex] is a solution to the nonhomogeneous equation. Show that [tex]u = a v + w[/tex] is a solution to the nonhomogeneous equation for any constant [tex]a[/tex].

Sagot :

GinnyAnswer
To show that [tex]\( u = a v + w \)[/tex] is a solution to the nonhomogeneous equation [tex]\( L(u) = f \)[/tex] for any constant [tex]\( a \)[/tex], we will follow these steps:

1. Understand the Given Information:
- [tex]\( L \)[/tex] is a linear operator.
- [tex]\( L(\alpha u + \beta v) = \alpha L(u) + \beta L(v) \)[/tex] for any scalars [tex]\(\alpha\)[/tex] and [tex]\(\beta\)[/tex].
- [tex]\( v \)[/tex] is a solution to the homogeneous equation, i.e., [tex]\( L(v) = 0 \)[/tex].
- [tex]\( w \)[/tex] is a solution to the nonhomogeneous equation, i.e., [tex]\( L(w) = f \)[/tex].

2. Express [tex]\( u \)[/tex] in Terms of [tex]\( v \)[/tex] and [tex]\( w \)[/tex]:
- Define [tex]\( u \)[/tex] as [tex]\( u = a v + w \)[/tex], where [tex]\( a \)[/tex] is a constant.

3. Apply the Linear Operator [tex]\( L \)[/tex] to [tex]\( u \)[/tex]:
- Calculate [tex]\( L(u) \)[/tex].

4. Use the Linearity Property:
- Use the property of the linear operator to express [tex]\( L(u) \)[/tex] in terms of [tex]\( L(v) \)[/tex] and [tex]\( L(w) \)[/tex].

5. Substitute the Known Values:
- Replace [tex]\( L(v) \)[/tex] with 0 and [tex]\( L(w) \)[/tex] with [tex]\( f \)[/tex].

6. Simplify the Expression:
- Show that the result equals [tex]\( f \)[/tex].

### Detailed Steps:

1. Apply [tex]\( L \)[/tex] to [tex]\( u \)[/tex]:
[tex]\[ L(u) = L(a v + w) \][/tex]

2. Use the Linearity Property of [tex]\( L \)[/tex]:
According to the property of the linear operator:
[tex]\[ L(a v + w) = a L(v) + L(w) \][/tex]

3. Substitute [tex]\( L(v) \)[/tex] and [tex]\( L(w) \)[/tex] with Their Known Values:
Since [tex]\( v \)[/tex] is a solution to the homogeneous equation, [tex]\( L(v) = 0 \)[/tex]. Also, since [tex]\( w \)[/tex] is a solution to the nonhomogeneous equation, [tex]\( L(w) = f \)[/tex]. Therefore:
[tex]\[ L(u) = a L(v) + L(w) = a \cdot 0 + f \][/tex]

4. Simplify the Expression:
[tex]\[ L(u) = 0 + f = f \][/tex]

### Conclusion:

Since [tex]\( L(u) = f \)[/tex], we have shown that [tex]\( u = a v + w \)[/tex] is indeed a solution to the nonhomogeneous equation [tex]\( L(u) = f \)[/tex] for any constant [tex]\( a \)[/tex].

Thus, we conclude that:
[tex]\[ u = a v + w \text{ is a solution to the nonhomogeneous equation } L(u) = f \text{ for any constant } a. \][/tex]
Thank you for your visit. We're dedicated to helping you find the information you need, whenever you need it. We hope our answers were useful. Return anytime for more information and answers to any other questions you have. Thank you for visiting Westonci.ca. Stay informed by coming back for more detailed answers.