Practice Basis Of An Eigenspace (32.2) - Basis of Eigenvectors - Mathematics (Civil Engineering -1)

Practice: Basis of an Eigenspace

Lesson 32.2Basis of Eigenvectors

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6 sample questions

Question 1.
What is an eigenvalue?
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Recall how it relates to linear transformations.
Question 2.
Define an eigenspace.
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Think about the vectors satisfying the eigenvalue equation.
3Question 3.
What is the eigenspace corresponding to an eigenvalue?
- A set of eigenvalues only
- A set of eigenvectors and the zero vector
- A matrix that represents linear transformation
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Think about which elements make up the eigenspace.
4Question 4.
True or False: The basis of an eigenspace is unique.
- True
- False
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Consider how many different ways you could represent the same space.
Question 5.
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.
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Do you remember how to find eigenvectors once the eigenvalues are known?
Question 6.
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.
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Recall the diagonalization process and why linearly independent eigenvectors matter.

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