The final exam will be on Thursday, Dec. 15, 5:30-7:30 p.m. in our usual room. The exam is closed book and closed notes. No calculators are permitted. The exam will cover the entire semester, but will place more weight on the following sections: 5.1-5.4, 6.1-6.5.

The design of the test will be slightly different from earlier tests--there will be more points for A and B level material.  But the scores will be scaled accordingly.  This will give you a better chance to improve your grade with a strong performance on the final. 

"C"-level material

Fundamental Skills and Concepts

1. ROW REDUCTION: Make absolutely sure you can do this flawlessly.
2. Find the general solution to a system of equations, a vector equation, or a matrix equation.
3. Write the solution to a linear system in parametric vector form.
4. Determine when a set of vectors is linearly independent or show that it is linearly dependent.
5. Determine if a given vector is in the span of a set of vectors.
6. Find a vector whose image under a matrix transformation is a given vector.
7. Determine whether a given matrix transformation is one-to-one or onto.
8. Know the properties of matrix multiplication (in particular, know what is different from multiplication of real numbers).
9. Determine the transpose of a matrix.
10. Determine when a matrix equation has more than just the "trivial" solution.
11. Multiply two matrices together.
12. Calculate the determinant of a two by two matrix.
13. Find the inverse of a two by two matrix.
14. THE INVERTIBLE MATRIX THEOREM !!!.
15. Calculate the determinant by cofactor expansion.
16. Relationship between determinant and invertibility.
17. Calculate the determinant of the product of two square matrices (given the determinant of each of the matrices).
18. Determine whether a set is a subspace of a vector space.
19. Determine the null & column space of a given matrix. (write the answer in parametric vector form).
20. Determine when a set of vectors forms a basis.
21. Determine the coordinates of a vector in R^n relative to a given basis.
22. Determine the rank of a matrix.
23. Given a vector, determine whether it is an eigenvector for a given matrix.
24. Express a subspace as the span of a given set of vectors.
25. Construct a basis from a spanning set by discarding unneeded vectors.
26. Construct a basis from a linearly independent set of vectors by adding vectors.
27. Find the dimension of a given vector space.
28. Find a basis for Nul A, Col A, and Row A.
    
New stuff (since the last exam)

29. Determine whether a given vector is an eigenvector for a matrix.
    
30. Determine whether a given scalar is an eigenvalue for a matrix, and find a basis for the associated eigenspace.
31. How does invertibility relate to the eigenvalues of a matrix?
32. Determine the characteristic equation for a matrix.
33. Solve the characteristic equation to find the eigenvalues of a matrix.
34. Determine the algebraic multiplicity of an eigenvalue.
35. Calculate the inner product (or dot product) of two vectors.
36. Determine the length (or norm) of a vector.
37. Find the distance between two vectors.
38. Determine when two vectors are orthogonal (by calculating their inner product).
39. Determine when a set is orthogonal.
40. Given an orthogonal basis for a subspace, write a given vector y as a linear combination of the vectors in this basis.
41. Given two vectors y and u, calculate the orthogonal projection of y onto u.
42. Find the component of y orthogonal to u.
43. Determine whether a given orthogonal set is orthonormal. If it isn't, scale the vectors in the set to create an orthonormal set.
44. Calculate the orthogonal projection of a vector onto a subspace.
45. Determine the best approximation within a subspace of a given vector not in the subspace.
46. Given a basis for a subspace, use the Gram-Schmidt process to determine an orthogonal or orthonormal basis.
47. Find a least squares solution to an inconsistent linear system Ax=b.

"B"-level material

Key skills and concepts

1. Find the standard matrix of a linear transformation.
2. Construct an elementary matrix corresponding to any given elementary row operation.
3. Give a geometric description of the span of a set of vectors in R^3 (be careful to check whether they are linearly independent).
4. Invert matrices using row-reduction.
5. Calculate the determinant of a matrix by row reduction.
6. Know what effect each of the elementary row operations has on the determinant.
7. Determine the kernel and range of a linear transformation.
8. Determine the change of coordinates matrix from a given basis to the standard basis.
   (new stuff)
   
9. Linear independence of eigenvectors corresponding to different eigenvalues.
10. Use the eigenvalues and eigenvectors of a matrix to diagonalize the matrix.
11. Understand the properties of inner products given by Theorem 1, page 370.
12. KNOW the relation between the orthogonal complement of row A and the nullspace of A. (SEE THEOREM 3, PAGE 381 !!!)

"A"-level material

1. Explain the relationship between matrix multiplication and composition of linear transformations.
2. Prove Theorem 8 on page 69.
3. Use the definition of a linear transformation to show that T(0) =0.
4. Explain why the standard matrix of a linear transformation is given by [T(e_1) T(e_2) ... T(e_n)].
5. Show that the inverse of a square matrix, if it exists, is unique.
6. Prove that a certain set along with corresponding operators is (or isn't) a vector space.
   
7. Prove certain properties about a vector space using only axioms of vector spaces.
   
8. Given the coordinates of a vector relative to a basis, determine the coordinates of the vector relative to a different basis.
9. Thoroughly understand the Invertible Matrix Theorem, including any additions to it covered in Chapters 3 and 4.
10. Determine the coordinates of a vector in a general vector space relative to a given basis.
    (New Stuff)
    
11. Explain why the roots of the characteristic equation give the eigenvalues of a matrix.
12. Calculate the nth power of a square matrix by first diagonalizing it.
13. Show that if U has orthonormal columns, then U'U=I (where U' is the transpose of U).
14. Calculate the QR factorization of a matrix A.
15. Use the QR factorization to determine a least squares solution to Ax=b.
    
16. Prove theorem 5 on page 385.
17. Do any of the "A"-level problems assigned in the study guides.