Practice Summary Table Of Concepts (32.14) - Basis of Eigenvectors

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6 sample questions

Question 1.
What is an eigenvector?
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Remember that `λ` is the eigenvalue.
Question 2.
Define eigenspace.
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Think about the vectors that satisfy `Av = λv`.
3Question 3.
What defines an eigenvector?
- A vector that changes direction under transformation.
- A linear combination of vectors.
- A non-zero vector satisfying Av = λv.
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Focus on the eigenvalue equation.
4Question 4.
True or False: The eigenvalue's algebraic multiplicity can be greater than its geometric multiplicity.
- True
- False
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Remember the definitions of both types of multiplicity.
Question 5.
Given a matrix A = [[2, 1], [1, 2]], find its eigenvalues and eigenvectors. Discuss if it is diagonalizable.
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Remember to use the determinant for the characteristic polynomial.
Question 6.
Consider a symmetric matrix B with eigenvalues 4, 4, and 2. Determine the geometric and algebraic multiplicities, and explain the implications for diagonalization.
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Think about how the multiplicities relate to the eigenspace dimensions.

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