Math205/Linear Algebra I Test Study Guide

----Final Exam-----

The final exam is scheduled as follows:

     (To Be Announced By The College)

    

....General Information About The Exam....

....(this information is the same as it was for Test #1 & #2)....

....(check the test #1 test guide page for details )....

A FEW IMPORTANT REMINDERS!

The test will be worth 120 points and will constitute 
between 33% & 38% of the total number of points
for the course.

You will have 2 hours to work on the test.

All of your work must be placed in the "Blue Book(s)"
that will be provided for you.

The answers in the Blue Book(s) must appear in the same order
as they appear on the test sheet, and so you will need to
plan ahead and leave necessary space if you intend to
work the problems out of order.

    

....Specific Information About The Exam....

   The first part of the final will be worth 40 points and will
   cover topics included on the first two tests. To prepare for
   this part of the exam you should revisit those exams since
   quesions on this part of the final will be similar to the 
   questions on those exams.
   
   The second part of the final will be worth 80 points and will 
   the following topics:


   0. All assigned homework problems:
         
   1. Be able to define the Rank of a matrix
   
   2. Be able to define the Nullity of a matrix
   
   3. Be able to define Linearly Independent vectors
   
   4. Be able to define Spanning Vectors
   
   5. Be able to define a Basis
          
   6. Be able to prove the pivot rows and columns of an Echelon matrix
      are Linearly Independent
      
   7. Be able to prove N(A)=N(U), where PA=LU
   
                        t 
   8. Be able to Prove A Y"dot"X = Y"dot"AX
   
                                                  t
   9. Be able to prove N(A) "is Orthogonal to" R(A ) 
    
                           t                        
   10. Be able to prove N(A ) "is Orthogonal to" R(A)   
      
   11. Be able to list all members of Sn (all permutations on
       n objects) using the notation used in class

   13. Be able to define an inversion of a permutation
   
   14. Be able to define even and odd permutations and determine 
       whether any given permutation is even or odd    
      
   15. Given an element of Sn,say "sigma", be able to determine 
       the "sgn" of sigma i.e. sgn(sigma) = (-1) raised to the number
       of inversions performed by sigma
      
   16. Define the determinant of an nxn matrix in terms of "Sigma" notation,
       permutations and "sgn"s of permutations

   17. Be able to describe the "cofactor" method of calculating 
       determinants
      
   18. Be able to show that the results of definition 16 applied
       to a 2x2 or 3x3 matrix is equivalent to the "Cofactor" method
       resulting from going along any row or column.
      
   19. Using the "cofactor" method, show that the determinant of any
       upper or lower triangular matrix is equal to the product of the 
       diagonal entries and the determinant of any matrix with a 
       column or row of zeros is equal to zero.
      
   19. Assuming that det(AB)=detAdetB, and the determinant of an 
       elementary matrix is not equal to zero (1,c or -1), prove 
       that an nxn matrix is invertible if and only if its 
       determinant is not zero.
   
   20. Be able to define the Eigenvalues and associated Eigenvectors
       of a matrix A.
   

    (last updated 2:00PM on 1-18-99)