32.3 - Steps to Find Basis of Eigenvectors
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Practice Questions
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What is the first step to find the basis of eigenvectors for a given matrix?
💡 Hint: Think about how you set up the determinant.
What does the eigenspace correspond to?
💡 Hint: Consider what defines a set of vectors for a given λ.
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Interactive Quizzes
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What do you solve to find eigenvalues for matrix A?
💡 Hint: What condition makes the determinant zero?
True or False: The eigenspace for an eigenvalue includes only the zero vector.
💡 Hint: Think about how many vectors make up an eigenspace.
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Challenge Problems
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Given the 2x2 matrix A = [[1, 2], [0, 1]], calculate its eigenvalues and corresponding eigenvectors, discussing the implications of your findings.
💡 Hint: What form will the characteristic polynomial take?
For the matrix B = [[4, 1], [0, 4]], derive the eigenvectors and explain the relevance of the result for a repetitive eigenvalue.
💡 Hint: How does the multiplicity of eigenvalues affect the linear independence of eigenvectors?
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