Math205/Linear Algebra I Test Study Guide-Spring 1999

----Test # 2-----

(last updated 10:00AM on 3-24-99)

Second Test:Tuesday, March 30, 1999

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

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

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

A FEW IMPORTANT REMINDERS!

The test will be worth 80 points and will constitute 
between 22% & 25% of the total number of points
for the course.

You will have 1hr and 20min 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 test will include only course material covered up to
and including Tuesday, March 23, 1999.

0. All assigned homework problems: Sections 2.3, 2.4, 2.5,
   2.6, 2.7, 3.1, 3.2, 3.4, 3.5 (skip prob #6) 

1. Assuming that both matrix A and B are invertible and of
   the same size, be able to prove that the inverse of AB is  
   B-inverse times A-inverse
   
2. Be able to explain why given a matrix A, which can be row
   reduced without swaps, then A = LU, where L is lower-triangular
   ,invertible, with ones on the diagonal and U is upper
   triangular (echelon)
   
3. Be able to show that if A=LU, where L is lower-triangular and
   with ones on the diagonal, and U is upper-triangular(echelon), then
   AX=B is solvable if and only if LY=B and UX=Y are solvable. Also,
   be able to explain why LY=B is always solvable, and under what 
   conditions UX=Y is solvable or not
   
4. Be able to define the transpose of a matrix   

5. Be able to prove that if A inverse exists, then A-transpose is
   also invertible and that the inverse of A-transpose is the 
   transpose of A-inverse

6. Be able to give a detailed definition of a Vector Space

7. Define a subspace of a vector space (2 ways)

8. Define the four fundamental subspaces associated with
   a matrix A
 
9. Be able to prove N(A) is a subspace of Rn and and R(A) ( C(A) )
   is a subspace of Rm

10. Be able to define the rank and nullity of a matrix A and
    give formulas for how these values are used to determine the
    dimensions of the four fundamental subspaces.
    
11. Given matrix A, be able to describe N(A) in terms of its
    "basis" vectors.
   

(last updated 10:00AM on 3-24-99)