32.2. Basis of an Eigenspace

Try these first

1. 1.

   What is an eigenvalue?

   Hint

   Recall how it relates to linear transformations.

2. 2.

   Define an eigenspace.

   Hint

   Think about the vectors satisfying the eigenvalue equation.

3. 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

   Hint

   Think about which elements make up the eigenspace.

4. 4.

   True or False: The basis of an eigenspace is unique.

   • True
   • False

   Hint

   Consider how many different ways you could represent the same space.

5. 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.

   Hint

   Do you remember how to find eigenvectors once the eigenvalues are known?

6. 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.

   Hint

   Recall the diagonalization process and why linearly independent eigenvectors matter.