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 is an matrix, and and are eigenvectors of corresponding to distinct eigenvalues and , respectively. This means:
Step 2: Assume Linear Dependence
Assume for contradiction that and are linearly dependent. If they are dependent, there exists a scalar such that:
Step 3: Substitute and Derive Contradiction
Using the eigenvalue-eigenvector relationship for , substitute into : Simplify: Divide through by :
This implies that is an eigenvector corresponding to , which contradicts the assumption that corresponds to with . Hence, and cannot be linearly dependent.
Step 4: Conclusion
Therefore, and are linearly independent.
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Relative Questions:
- What are the implications of eigenvector independence in solving systems of differential equations?
- Can eigenvectors corresponding to the same eigenvalue ever be linearly dependent?
- How does the proof of independence extend to more than two eigenvectors with distinct eigenvalues?
- What role does the spectral theorem play in this proof?
- 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
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