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Practice Questions
Test your understanding with targeted questions related to the topic.
Question 1
Easy
Define an eigenvector.
💡 Hint: Think about the relationship between **A**, **v**, and **λ**.
Question 2
Easy
What is the formula for determining the eigenspace associated with an eigenvalue?
💡 Hint: Consider the equation form involving the matrix **A**.
Practice 4 more questions and get performance evaluation
Interactive Quizzes
Engage in quick quizzes to reinforce what you've learned and check your comprehension.
Question 1
What defines an eigenvector?
- A vector that can only be zero
- A vector that satisfies Av = λv
- Any random vector
💡 Hint: Focus on the defining equation involving the matrix **A**.
Question 2
True or False: The eigenspace of an eigenvalue consists solely of the eigenvector itself.
- True
- False
💡 Hint: Consider whether the eigenspace could have multiple vectors.
Solve 1 more question and get performance evaluation
Challenge Problems
Push your limits with challenges.
Question 1
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.
Question 2
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.
Challenge and get performance evaluation