1. For the matrix
   How does this matrix act on a unit square for k=0, k=1, k=2?
   What are the eigenvalues and eigenvectors for k=0, k=1, k=2?
   What values of k result in the matrix Sv having just 1 linearly independent eigenvector.
   What values of k result in the matrix Sv having only 1 eigenvector.
   What values of k result in the matrix Sv having infinitely many eigenvectors?
   What values of k result in the matrix Sv having infinitely many linearly independent eigenvectors?
   What values of k result in the matrix Sv having eigenvectors that form a basis for R²?
2. Give an example of a Fox-Rabbit state matrix where the foxes and rabbits always die off. In what directions do they die off for various a1 and a2 coefficients in the decomposition equation?
3. Give an example of a Fox-Rabbit state matrix where the foxes and rabbits sometimes die off, but in other situations they stabilize. In what directions do they do this for various a1 and a2 coefficients in the decomposition equation? ?
4. Give an example of a Fox-Rabbit state matrix where the foxes and rabbits sometimes die off, but in other situations they grow. In what directions do they do this for various a1 and a2 coefficients in the decomposition equation?
5. If the Determinant of a matrix A is 0, how many solutions does the system Ax=b have?
6. If the Determinant of a matrix A is nonzero, how many solutions does the system Ax=b have?
7. If the Determinant of a matrix A is 0, how many solutions does the system Ax=0 have? Why? Could the column vectors of A span Rⁿ? Why or why not?
8. Is x+y = 1 a subspace in R²?
   If so, then provide a basis and specify what geometric object this is.
   If not, then first prove why using axiom 1 or axiom 6 from the vector space axioms. Next create a subspace that is similar to the original set, specify what is similar and what is different, and provide a basis for the subspace.
9. Look at the vectors (1,2,3), (4,5,6), (5,7,9), and (-3,-3,-3).
   Part A Are these vectors linearly independent in R³?
   Part B Do these vectors span R³? If not, add a vector so that the vectors will span R³.
   Part C Use the system of equations that is used in the definition of linearly independent and solve them for solutions. Do the solutions form a vector space? Why or why not? If so, what is a basis?
10. If a matrix is Gaussian Eliminated to
    Does the system every have 0 solutions? If so, for what k?
    Does the system ever have infinitely many solutions? If so, for what k, and then write out the solutions in terms of x_p and x_h
    Does the system ever have 1 solution? If so, for what k?
11. What is the practicle applications of span and li in the cement mixing problem? What is an example of a mix with all positive entries that can be formed mathematically from the 5 basic mixes in your PS 5 problem set, but cannot be formed physically?