Math205/Linear Algebra I Test Study Guide
----Final Exam-----
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(last updated 9:30AM on 4-27-99)
The final exam is scheduled as follows:
Math205-02..Thursday, May 6, 1999, 4:40-6:40PM in Holman Hall 204
Math205-03..Tuesday, May 4, 1999, 6:50-8:50PM in Holman Hall 204
....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.
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 the questions 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: Sections 3.5 (prob #6), 3.6,
4.1, 4.2, 5.1, 5.2, 6.1,
1. Be able to define Linearly Independent vectors
2. Be able to define Spanning Vectors
3. Be able to define a Basis
4. Be able to define the dimension of a vector spave
5. Be able to explain using some math, some pictures and some English why
the pivot rows and columns of an Echelon matrix are Linearly Independent
6. Be able to prove N(A)=N(U), where PA=LU and P is an invertible permutation
matrix, L is a square, lower-triangular matrix with 1's on the diagonal, and U
is upper Echelon.
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7. Assuming AX dot Y = X dot A Y, be able to prove N(A) "is Orthogonal to" R(A )
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, and N(A ) "is Orthogonal to" R(A)
8. Be able to list all members of Sn (all permutations on
n objects) using the notations used in class for n <= 4
9. Be able to define an "inversion" for a permutation
10. Be able to define even and odd permutations and determine
whether any given permutation in Sn is even or odd for n <= 4
11. 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 (for n <= 4)
12. Define the determinant of an nxn matrix in terms of "Sigma" notation,
permutations in Sn and "sgn"s of permutations
13. Be able to describe the "cofactor" method of calculating
determinants (using minors and cofacotrs)
14. Be able to show that the results of definition 12 applied
to a 2x2 or 3x3 matrix is equivalent to the "Cofactor" method
resulting from going along any row or column
15. 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.
16. Assuming that det(AB)=detAdetB, and that 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
17. Be able to define the Eigenvalues and associated Eigenvectors
of a matrix A
18. Be able to define the Proj b, where a and b line in Rn
a
19. Be able to define the Projection of b (for b in Rm) onto the column space
of an mxn matrix A, and say under what conditions is the solution to the
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equation A Ax = A b unique
(last updated 9:30AM on 4-27-99)