32 - Basis of Eigenvectors
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Practice Questions
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Define an eigenvector.
💡 Hint: Think about the relationship between **A**, **v**, and **λ**.
What is the formula for determining the eigenspace associated with an eigenvalue?
💡 Hint: Consider the equation form involving the matrix **A**.
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Interactive Quizzes
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What defines an eigenvector?
💡 Hint: Focus on the defining equation involving the matrix **A**.
True or False: The eigenspace of an eigenvalue consists solely of the eigenvector itself.
💡 Hint: Consider whether the eigenspace could have multiple vectors.
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Challenge Problems
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Consider the matrix B = [[5, 4], [2, 3]]. Compute its eigenvalues and eigenvectors and discuss their significance in terms of modulating the behavior of a system.
💡 Hint: Set up the characteristic equation and solve for λ.
A matrix has three eigenvalues: 2, 2, and 3. What can you say about its diagonalizability based on geometric and algebraic multiplicities?
💡 Hint: Realize the importance of the corresponding eigenspaces.
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