Practice - Properties of Eigenvectors
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Practice Questions
Test your understanding with targeted questions
What happens to an eigenvector if it is multiplied by a scalar?
💡 Hint: Think about if it stretches or shrinks.
If a matrix has 3 distinct eigenvalues, how many linearly independent eigenvectors will it have?
💡 Hint: Remember the definition of distinct eigenvalues.
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Interactive Quizzes
Quick quizzes to reinforce your learning
What does it mean if eigenvectors are linearly independent?
💡 Hint: Recall the definition of linear independence.
True or False: All eigenvectors of symmetric matrices are complex.
💡 Hint: Think about the properties of symmetric matrices.
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Challenge Problems
Push your limits with advanced challenges
Given a 3x3 real symmetric matrix, prove that its eigenvectors corresponding to distinct eigenvalues are orthogonal.
💡 Hint: Apply the eigenvalue equation and manipulate using the symmetry property.
Consider a 4x4 matrix that has 4 distinct eigenvalues. How would you go about finding its eigenvectors? Discuss the method and implications for usage in scenarios such as structural analysis.
💡 Hint: Remember to check for linear independence.
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