1. Which of the following must be true if A, a 2x2 matrix, has a nonzero eigenvector a satisfying Aa= 5a?
   a) A's eigenvectors are all of R²
   b) A must have at least an entire line through the origin in R² as it's eigenvectors, where the vectors get stretched by 5.
   c) A can have just the one eigenvector a that is stretched by 5.
   d) A has two eigenvectors, the second being from Maple
   e) none of the above
   
   
   
   
2. If the reduced augmented matrix for the system (A-lambdaI)x=0 is
   Matrix([[0,0,0],[0,0,0]]),
   with A as a 2x2 matrix
   then the non-zero (real) eigenvectors of A are:
   a) none
   b) a line through the origin
   c) all of R²
   d) a subspace of R³ (with 3 coordinates)
   e) none of the above
   
   
   
   
   
   
3. What are the non-zero real eigenvectors of
   Matrix([[cos(Pi),-sin(Pi)],[sin(Pi),cos(Pi)]])?
   a) none
   b) a line of eigenvectors
   c) two different lines of eigenvectors
   d) all of R²
   e) none of the above
   
   
   
   
4. If we write a basis for the eigenspace of
   Matrix([[cos(Pi),-sin(Pi)],[sin(Pi),cos(Pi)]]), how many vectors does it have?
   a) 0
   b) 1
   c) 2
   d) infinite
   e) none of the above
   
   
   
   
5. What are the non-zero real eigenvectors of
   Matrix([[cos(Pi/5),-sin(Pi/5)],[sin(Pi/5),cos(Pi/5)]])?
   a) none
   b) a line of eigenvectors
   c) two different lines of eigenvectors
   d) all of R²
   e) none of the above
   
   
   Solutions
   1. b)
   2. c)
   3. d)
   4. c)
   5. a)