Discover answers to your questions with Westonci.ca, the leading Q&A platform that connects you with knowledgeable experts. Join our Q&A platform to connect with experts dedicated to providing precise answers to your questions in different areas. Explore comprehensive solutions to your questions from knowledgeable professionals across various fields on our 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 :

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]