Practice Summary Of Key Concepts (32.9) - Basis of Eigenvectors

Practise this section

9 questions

Flashcard drill
4 cards from this section. Quick, good the night before a test.
Exercise
6 questions · about 4 min · pass mark 70%. Hints available.
Sign up to start the exercise
Quiz
3 questions · about 2 min · pass mark 70%. Hints available.
Sign up to start the quiz

6 sample questions

Question 1.
Define an eigenspace in your own words.
Show hintHide hint
Think about what null space means in context.
Question 2.
What does it mean for a matrix to be diagonalizable?
Show hintHide hint
Focus on how matrices can be expressed.
3Question 3.
What represents the null space of (A - λI)?
- Eigenvectors
- Eigenvalues
- Eigenspaces
Show hintHide hint
Consider the definition of eigenspaces.
4Question 4.
True or False: A matrix can be diagonalizable with fewer than n eigenvectors.
- True
- False
Show hintHide hint
Reflect on the definition of diagonalizable matrices.
Question 5.
Given the matrix A = [[2, 0], [0, 3]], find its eigenvalues and check if A is diagonalizable.
Show hintHide hint
Calculate the eigenvalues and check the linear independence of the eigenvectors.
Question 6.
Consider a 3x3 matrix with eigenvalues of 1, 1, and 2. Determine if it's diagonalizable.
Show hintHide hint
Check the geometric vs algebraic multiplicity for eigenvalues.

5 more questions in this section. Enroll free to practise them all.