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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]
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