Practice Eigenvalues And Eigenvectors Of Linear Transformations (28.12)

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6 sample questions

Question 1.
Define an eigenvector in your own words.
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Think about how a vector behaves during scaling.
Question 2.
What is the relationship between eigenvalues and eigenvectors?
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Consider the equation T(v) = λv.
3Question 3.
What is an eigenvalue?
- A vector that defines direction
- A scalar that represents scaling
- The transformation itself
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Think about what is left unchanged in the direction of an eigenvector.
4Question 4.
True or False: Eigenvectors can be the zero vector.
- True
- False
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Remember the definition of eigenvectors and their importance.
Question 5.
Consider a matrix A = [[3, 1], [0, 2]]. Calculate the eigenvalues and eigenvectors.
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Start by plugging λ back into (A − λI).
Question 6.
Explore a real-world scenario where an engineer must consider the eigenvalues of a structure to ensure stability. Explain the implications of the eigenvalues on design choices.
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Think about what happens if the natural frequency matches an external force frequency.

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