Math205/Linear Algebra I Test Study Guide-Spring 1999
-----Test # 1------
First Test:Tuesday, February 23, 1999
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....General Information About The Exam....
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.
Your answers will be graded on the basis of correctness,
completeness, and neatness.
The best philosophy you can have during the exam is to
leave as little to the imagination as is possible. The basic
rule is that you write down only things that are true and
give reasons as to why they are true.
The point value of each question will be noted for each
question. Take this into consideration while managing
your time during the exam. Don't spend 50% of your time
during the examination answering questions that count
for only 20% of the total points.
Partial credit will be awarded and so you should try to do
some work on as many questions as you can. More weight will
be placed on your ability to demonstrate a correct process
that may be followed to arrive at the correct answer than
the final answer itself. Therefore, a correct final answer
with no supporting work will receive little to no credit,
and a correct process that contains some minor error(s)
that caused an incorrect final answer will receive up to
100% of the full credit, depending on the extent of the
error(s).
Do not feel bad if you are unable to finish the exam.
Letter grade assignment for the test will be based
on a curve with cut-offs less than or equal to straight
percentage, and so it is possible to earn a good grade
without completing the answers to all of the questions.
Talk to your friends after the exam. The amount of work
that others were able to complete will be a good
indicator of your situation.
....Specific Information About The Exam....
The test will include only course material covered up to
and including Tuesday, February 16, 1999.
Assigned homework problems will form the foundation for many
of the test questions. Some exam will either be taken directly
from the homework or be very similar to homework problems.
Other test questions will either be drawn from class notes
or directly from the text.
The best preparation for the exam is to practice working
problems and read over your class notes for understanding.
Remember that the more problems you work and the more
you study your notes, the better prepared you will be for
the exam. You are encouraged to attempt even more problems
than were placed on the Assignment Web Page.
In addition to all assigned problems that are listed
on the web (or problems similar to them) the
following specific topics should be studied:
1. Be able to state and illustrate all of the properties
satisfied by vector operations such as "addition",
"scalar multiplication", "length or norm", and "dot or inner
product" of vectors (ie properties including associative,
commutative,distributive, identity, inverse, etc )
2
2. Be able to prove (as done on the board in R ) that u"dot"v =
||u|| ||v|| cos (theta) , where theta is the angle between u & v.
3. Use 2 to prove |u"dot"v| <= ||u|| ||v||
4. define perpendicular vectors
5. prove two vectors are perpendicular if and only if their
dot product equals 0
6. Be able to write down the three equivalent ways of viewing
a system of m "Linear" equations in n unknowns. (ie row form,
column form and matrix equation form) and explain the geometric and/or
algebraic interpretation of each
7. Be able to define matrix addition and multiplication and state
and illustrate all of the properties matrices satisfy with
respect to these operations (ie assoc, commut, ident, inv ,etc)
8. Define the three types of elementary matrices and describe their
inverses; and describe what effect multiplying a matrix A on the
left or by elementary matrix has on the matrix A
9. Given an example such as:
3x +4y -11z = 4
6x -2y +2z = 9
then using either elementary equation operations (i.e. multiply
any equation by a non-zero constant, multiply one equation nby
a constatn and add it to another equation, or swap the order
of two equations), sthen olve the problem in such a way that
| x |
| y | can be expressed as the sum of a constant vector that
| z | solves the problem and the sum of scalar multiples
of vectors that solve AX=0
10. Given square matrix A, such as | 1 2 3 |
| 4 5 6 |
| 7 8 9 | -1
be able to use the Gauss-Jordan method to find A in two ways.
The first way is by elementary row operations on the matrix, and
the second way by using multiplication of A on the left by
corresponding elementary matrices.
11. Be able to prove that if there exist elementary matrices E , E ...E
1 2 k
such that (E E ...E E )A = I , then A(E E ...E E ) = I
k k-1 2 1 k k-1 2 1
, that is, be able to prove that a matrix has a left inverse if
and only if it has a right inverse.
(last updated 1:15PM on 2-17-99)