Practice Eigenvectors And Eigenspaces (32.1) - Basis of Eigenvectors

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6 sample questions

Question 1.
Define what an eigenvector is.
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Think about how it relates to matrices and transformations.
Question 2.
What does the eigenspace correspond to?
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Remember this definition involves the concept of null spaces.
3Question 3.
What is an eigenvector?
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Think of the relationship between the vector v and the transformation A.
4Question 4.
An eigenvalue can have multiple corresponding eigenvectors. True or False?
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Recall the definition of eigenspaces.
Question 5.
Given the matrix A = [[2, -1], [1, 1]], find its eigenvalues and eigenvectors. Discuss the implications of your results on the matrix's diagonalizability.
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Focus on the steps to derive the characteristic polynomial and the eigenvectors from there.
Question 6.
Explore a 3x3 matrix with one eigenvalue repeated three times but only one linearly independent eigenvector. How would you illustrate this with an example, and what does it signify?
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Remember to check both the characteristic polynomial and null space.

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