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]
