Math Problem Statement

show that two eigenvectors are linearly independant

Solution

To show that two eigenvectors corresponding to distinct eigenvalues are linearly independent, consider the following steps:

Step 1: Eigenvalue-Eigenvector Relationship

Suppose AA is an n×nn \times n matrix, and v1v_1 and v2v_2 are eigenvectors of AA corresponding to distinct eigenvalues λ1\lambda_1 and λ2\lambda_2, respectively. This means: Av1=λ1v1andAv2=λ2v2.A v_1 = \lambda_1 v_1 \quad \text{and} \quad A v_2 = \lambda_2 v_2.

Step 2: Assume Linear Dependence

Assume for contradiction that v1v_1 and v2v_2 are linearly dependent. If they are dependent, there exists a scalar cc such that: v1=cv2.v_1 = c v_2.

Step 3: Substitute and Derive Contradiction

Using the eigenvalue-eigenvector relationship for v1v_1, substitute v1=cv2v_1 = c v_2 into Av1=λ1v1A v_1 = \lambda_1 v_1: A(cv2)=λ1(cv2).A (c v_2) = \lambda_1 (c v_2). Simplify: cAv2=cλ1v2.c A v_2 = c \lambda_1 v_2. Divide through by c0c \neq 0: Av2=λ1v2.A v_2 = \lambda_1 v_2.

This implies that v2v_2 is an eigenvector corresponding to λ1\lambda_1, which contradicts the assumption that v2v_2 corresponds to λ2\lambda_2 with λ1λ2\lambda_1 \neq \lambda_2. Hence, v1v_1 and v2v_2 cannot be linearly dependent.

Step 4: Conclusion

Therefore, v1v_1 and v2v_2 are linearly independent.


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Relative Questions:

  1. What are the implications of eigenvector independence in solving systems of differential equations?
  2. Can eigenvectors corresponding to the same eigenvalue ever be linearly dependent?
  3. How does the proof of independence extend to more than two eigenvectors with distinct eigenvalues?
  4. What role does the spectral theorem play in this proof?
  5. Can eigenvectors of defective matrices always form a basis?

Tip:

When solving problems involving eigenvalues and eigenvectors, always check if the eigenvalues are distinct. If so, the corresponding eigenvectors are guaranteed to be linearly independent.

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Math Problem Analysis

Mathematical Concepts

Linear Algebra
Eigenvectors
Eigenvalues
Linear Independence

Formulas

A v1 = λ1 v1
A v2 = λ2 v2

Theorems

Theorem on Linear Independence of Eigenvectors Corresponding to Distinct Eigenvalues

Suitable Grade Level

Grades 11-12