In this exercise, we are tasked with proving that the function $L: V_1 \to V_1$, defined by $L(f) = f' + f$, is a linear mapping between real vector spaces. Let’s analyze the problem and tackle the question step-by-step.

Problem Breakdown:

1. Space Definition:

   • $C_\infty(\mathbb{R})$ represents the real vector space of all functions from $\mathbb{R}$ to $\mathbb{R}$ that can be differentiated any number of times.
   • $V_1$ is a subspace of $C_\infty$ generated by the functions $e^t$, $e^{-t}$, $\cos(t)$, and $\sin(t)$. These functions form a basis for $V_1$ and are linearly independent.

2. Function Definition:

   • The mapping $L$ is defined as $L(f) = f' + f$, where $f'$ denotes the derivative of $f$.

3. Goal:

   • To show that $L$ is a linear mapping. For a function to be linear, it must satisfy the following properties for any functions $f, g \in V_1$ and any scalar $c \in \mathbb{R}$:
     • Additivity: $L(f + g) = L(f) + L(g)$
     • Homogeneity: $L(c \cdot f) = c \cdot L(f)$

Solution Steps:

1. Additivity Check: $L(f + g) = (f + g)' + (f + g)$ Using the linearity of differentiation: $= f' + g' + f + g = (f' + f) + (g' + g) = L(f) + L(g)$ Therefore, $L(f + g) = L(f) + L(g)$, satisfying additivity.

2. Homogeneity Check: $L(c \cdot f) = (c \cdot f)' + c \cdot f$ Again, applying the property of differentiation with a constant: $= c \cdot f' + c \cdot f = c \cdot (f' + f) = c \cdot L(f)$ Thus, $L(c \cdot f) = c \cdot L(f)$, satisfying homogeneity.

Conclusion:

Since both properties (additivity and homogeneity) hold, $L$ is a linear mapping between real vector spaces.

Would you like a deeper explanation of each property or have questions about other aspects of this proof?

Related Questions:

1. How would we find the matrix representation of $L$ with respect to the basis $\{ e^t, e^{-t}, \cos(t), \sin(t) \}$?
2. What would happen to $L$ if we changed the basis of $V_1$?
3. Can we determine the eigenvalues and eigenvectors of $L$ in $V_1$?
4. How can we interpret the linear map $L$ geometrically in $V_1$?
5. How would this approach change if $L(f)$ were defined as $f' - f$ instead?

Tip:

When proving linearity, remember to check both additivity and homogeneity as they ensure the function respects vector space structure.