Math Problem Statement

brug f1 og f2

Vis at funktionen L er en lineær afbildning mellem reelle vektorrum.

Solution

To demonstrate that the function LL is a linear mapping between real vector spaces, we need to verify two properties of linearity for LL:

  1. Additivity: L(f+g)=L(f)+L(g)L(f + g) = L(f) + L(g)
  2. Homogeneity: L(cf)=cL(f)L(c \cdot f) = c \cdot L(f), where cc is a scalar.

Given:

  • The function L:V1V1L: V_1 \to V_1 is defined by L(f)=f+fL(f) = f' + f, where fV1f \in V_1.

Step 1: Check Additivity

For functions f,gV1f, g \in V_1: L(f+g)=(f+g)+(f+g).L(f + g) = (f + g)' + (f + g). Using the properties of derivatives and addition in functions: (f+g)+(f+g)=f+g+f+g=(f+f)+(g+g)=L(f)+L(g).(f + g)' + (f + g) = f' + g' + f + g = (f' + f) + (g' + g) = L(f) + L(g). Thus, additivity is satisfied: L(f+g)=L(f)+L(g)L(f + g) = L(f) + L(g).

Step 2: Check Homogeneity

For a scalar cRc \in \mathbb{R} and a function fV1f \in V_1: L(cf)=(cf)+(cf).L(c \cdot f) = (c \cdot f)' + (c \cdot f). Using the derivative property of scalar multiples: (cf)+(cf)=cf+cf=c(f+f)=cL(f).(c \cdot f)' + (c \cdot f) = c \cdot f' + c \cdot f = c \cdot (f' + f) = c \cdot L(f). Thus, homogeneity is satisfied: L(cf)=cL(f)L(c \cdot f) = c \cdot L(f).

Conclusion

Since LL satisfies both additivity and homogeneity, LL is indeed a linear mapping.

Would you like further details on this solution or any related questions?

Here are 5 follow-up questions that might help deepen understanding:

  1. How can we find the matrix representation of LL with respect to the basis {et,et,cos(t),sin(t)}\{e^t, e^{-t}, \cos(t), \sin(t)\}?
  2. What role do the basis functions et,et,cos(t),sin(t)e^t, e^{-t}, \cos(t), \sin(t) play in analyzing the behavior of LL?
  3. Could we determine the eigenvalues and eigenvectors of LL in V1V_1?
  4. What would the effect be if we changed the form of LL to L(f)=ffL(f) = f' - f instead?
  5. How does the concept of linearity help simplify operations with differential operators like LL?

Tip: Remember, verifying linearity involves proving both additivity and homogeneity, which are fundamental properties in vector space transformations.

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Linear transformations
Differentiation
Vector spaces
Linearity

Formulas

L(f) = f' + f
L(f + g) = L(f) + L(g)
L(c ⋅ f) = c ⋅ L(f)

Theorems

Linearity properties (Additivity and Homogeneity)

Suitable Grade Level

Undergraduate Mathematics