Math205/Linear Algebra I Test Study Guide-Fall 1998
-----Test # 1------
First Test:Friday, October 2, 1998
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....General Information About The Exam....
The test will be worth 80 points and will
constitute roughly 25% of the total number of test 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.
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 Friday, September 27, 1998.
Assigned homework problems will form the foundation for many
of the test questions. As much as 75% of the exam may 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, 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 assignment (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 matrices has on the matrix A
9. Define and invertible matrix and prove that the product of invertible
matrices is invertible and the inverse of the product is the product
of their inverses in the opposite order.
10. Be able to prove that if C is an invertible matrix then X is a solution
of AX=B if and only if x is a solution of (CA)X=CB. Also, be able to
describe what effect this matrix theroem has on the planar version of
the problem.
11. Given an example such as:
3x +4y -11z = 4 or | 3 4 -11 | | x | | 4 |
| | | | | |
6x -2y +2z = 9 | 6 -2 2 | | y | = | 9 |
| |
| z |
then using either elementary matrix operations on the matrix
version of the problem, or by the eqivalent operations on the
equations of the planes, solve 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
(last updated 1:45PM on 9-25-98)