This file will contain the review sheets for the two midterm exams 
     and the final exam for this course.

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     EXAM 1 REVIEW SHEET,  MATH 442,  SPRING 2000

     The exam will be Wed Feb 16 and will cover chapters 1 to 3.
     It will be open text.  One 8.5 by 11 inch crib sheet will be
     allowed, both sides.  Most of the questions will relate to examples
     rather than abstract proofs.

     POSSIBLE TYPES OF QUESTIONS:

     1. Is a given set V a vector space under given addition and scalar
     multiplication?

     2. Is U a subspace of V?

     3. Given subspaces U and W, find U+W.

     4. Can U + W be written as U (+) W?  [(+) means circle+.]

     5. Is a given list of vectors ( v(1), ..., v(n) ) independent?

     6. Find a basis of a given V.S. or subspace.

     7. Find the dimension of a given V.S. or subspace.

     8. Given a linear map T:V->W, find null T and range T and their
     dimensions.

     9. Find the matrix of a vector v or of a linear map T or of Tv 
     with respect to given bases.  Do some matrix algebra.  

     10. Is a given map T:V->W linear? injective? surjective? 1-1? onto?

     11. Is a given linear map invertible?  If so find a formula for
     its inverse.  

     12. Some True/False questions.  Example: If T:R^n -> R^n is linear
     and injective must it also be surjective?  Why?

     13. Prove a simple proposition, not one in the text.

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     EXAM 2 REVIEW SHEET,  MATH 442,  SPRING 2000

     The exam will be on Fri., Mar. 31 and will cover chapters 4, 5
     and 6 through the top of page 111.  It will be open text and in
     addition one 8.5 by 11 inch crib sheet will be allowed, both sides.
     Most of the questions will involve examples rather than abstract
     proofs.

     POSSIBLE TYPES OF QUESTIONS:

        CHAPTER 4:

     1. There might be a question about polynomials, somehow related 
     to exercises 4.1-4.5.

        CHAPTER 5:

     2. Find the eigenvalues of a given operator T.

     3. Find the eigenspace (or a basis for the eigenspace) of an
     operator T corresponding to some eigenvalue of T.

     4. Find the eigenvalues and eigenvectors of p(T), where p is some
     polynomial, given the eigenvalues and eigenvectors of T.

     5. Find eigenvalues of some simple operator which has a triangular
     matrix.

     6. There might be a question about a projection operator P = P_{U,W}.
     where V = U (+) W = the direct sum of U and W.  For example: find
     Pv for some given v in V.

     7. Determine whether a given operator T has a diagonal matrix.

        CHAPTER 6:

     8.  Calculate some inner products and norms and use them for some
     application such as orthogonality, etc.

     9.  Given vectors u and v in an inner product space, find the
     orthogonal projection of u onto v, or write u as a sum of vectors
     parallel and orthogonal to v.

     10. Use the Schwarz or triangle inequality.

     11. Use the Gram-Schmidt Orthogonalization Process to find an
     orthogonal or orthonormal basis for a subspace.

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     REVIEW SHEET FOR FINAL EXAM, MATH 442, SPRING 2000

     For the final exam you may use your text and notes.
     The types of questions will be chosen from the following list 
     (where R^n = n dimensional row or column space and P_n = the
     space of real polynomials of degree n or less):  

     1. Find a basis and the dimension of a subspace of R^n.

     2. Find a basis and the dimension of a subspace of P_n.

     3. Given some independent vectors v1, v2, v3, find an orthogonal
     basis for the subspace spanned by them, using the Gram-Schmidt
     process.

     4. Find eigenvalues and eigenvectors of a simple 2x2 or 3x3 matrix.

     5. Evaluate a 4 by 4 determinant using elementary row or column
     operations.

     6. Find a basis of the row space of an mxn matrix A by writing it
     in row reduced form.

     7. Find the inverse of a 2x2 or 3x3 matrix.

     8. Find some generalized eigenvectors of a simple 2x2 or 3x3 matrix.

     9. Verify the Cayley-Hamilton theorem for a given matrix.

     10. Show that similar matrices have the same eigenvalues.  

     11. Given that A = S D S^(-1), where D is diagonal, show that the
     columns of S are eigenvectors of A.

     12. Given a matrix in Jordan canonical form, find its minimum
     polynomial.
     
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ACTUAL MATH 442 FINAL EXAM QUESTIONS, SPRING 2000

   1) Given a linear functional find the dimension of its range 
and null spaces and find a basis for its null space.
   2) Find an orthogonal basis for a subspace of R^n (for some n) 
using the Gram-Schmidt process.
   3) For a simple 3x3 matrix find its eigenvalues and the 2nd 
level generalized eigenvectors.
   4) Evaluate a 4x4 determinant using elementary row ops to first 
reduce it to a 2x2.
   5) Find a basis and the dimension of the row space of a 4x4 
matrix.
   6) Verify that a given 2x2 matrix satisfies the Caley-Hamilton 
Theorem.
   7) Given the Jordan form of a 7x7 matrix A, find the number of 
independent eigenvectors of A, the characteristic polynomial of A, 
and the minimum polynomial of A.
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