32.2 - Basis of an Eigenspace
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Practice Questions
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What is an eigenvalue?
💡 Hint: Recall how it relates to linear transformations.
Define an eigenspace.
💡 Hint: Think about the vectors satisfying the eigenvalue equation.
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Interactive Quizzes
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What is the eigenspace corresponding to an eigenvalue?
💡 Hint: Think about which elements make up the eigenspace.
True or False: The basis of an eigenspace is unique.
💡 Hint: Consider how many different ways you could represent the same space.
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Challenge Problems
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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?
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.
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