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Practice Questions
Test your understanding with targeted questions related to the topic.
Question 1
Easy
What is an eigenvalue?
💡 Hint: Recall how it relates to linear transformations.
Question 2
Easy
Define an eigenspace.
💡 Hint: Think about the vectors satisfying the eigenvalue equation.
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 is the eigenspace corresponding to an eigenvalue?
- A set of eigenvalues only
- A set of eigenvectors and the zero vector
- A matrix that represents linear transformation
💡 Hint: Think about which elements make up the eigenspace.
Question 2
True or False: The basis of an eigenspace is unique.
- True
- False
💡 Hint: Consider how many different ways you could represent the same space.
Solve 1 more question and get performance evaluation
Challenge Problems
Push your limits with challenges.
Question 1
Given a 2x2 matrix \( A = \begin{bmatrix} 3 & 1 \ 0 & 2 \end{bmatrix} \) find the eigenvalues, determine the eigenspaces, and describe the basis of each eigenspace.
💡 Hint: Do you remember how to find eigenvectors once the eigenvalues are known?
Question 2
Consider a symmetric matrix for which you have found eigenvalues of 1, 2, and 3 with linearly independent eigenvectors. Explain how these can be used for diagonalization.
💡 Hint: Recall the diagonalization process and why linearly independent eigenvectors matter.
Challenge and get performance evaluation