Final Exam
MA 2071
This course has
attempted to study traditional topics from Linear Algebra in class and then, by
way of group projects, use some of this material for applications appropriate
to the major.
The areas of core
material we have covered may be organized into 5 areas:
1) Solutions of Linear Systems
homogeneous
& particular solutions
3 basic types of
solutions
linearity
may
be viewed in column terms
2) Matrix Algebra & Arithmetic
computations
and matrix arithmetic
symbolic
manipulation of matrices
linearity
of matrix multiplication
inverses
of square matrices
powers
of matrices (see diagonalization later)
3) Linear
Transformations
special
types of functions
defined
entirely by their effect on a basis for the domain
directly related to definition
relate
easily to differential equations due to linearity of derivatives
4) Vector
Spaces
have
two closure properties
built
from a basis; coordinates provide details
may
be finite or infinite dimensional (Rn or spaces of functions)
appropriate
domain, range for linear transformations
5)
Diagonalization
basic
definitions key
finding
optimal coordinate system to simplify the matrix of a linear transformation
considerably
useful
for powers of matrices, graphics and quadratic functions
Different parts of this material are
useful for different majors;
some are used very computationally while others are more
important from a conceptual point of view.
The most common concept to thread through all of this is clearly that of
linearity – the ability to take
things apart and put them together.
Your final exam is to take two
of these areas and discuss and describe how they relate to some portion of your
major. Your discussion
should not have any computations or derivations
in it (we've done that)
(phrases such as "it can be shown that…." or
"a derivation would show that … " are good)
should clearly refer to important concepts
should have precise and appropriate use of mathematical
symbols and terminology
should make use of, or refer to, 1 or 2 illustrative
problems in your major as focal points
mention the appropriate use of technology (software and
other) in your work
generally indicate that you have thought about the relation of linear algebra to mathematics
applicable to
your major.
It should take you
several pages and a good part of the 50 minute class to develop an
appropriately well developed response.
A
special note to CS majors. One
can take two views of you: having to do
with computers or
having to do with computer science. The
former could mean almost anything today.
The latter literally may be taken to refer to the science of algorithms and is what we are
referring to here. Your job on the final
is a little different. Start with a
definition of algorithm. Take both of
projects you worked on and discuss and summarize the algorithms used in them. How did you decide on them? Be sure to mention
any attempts you made to increase the efficiency of your code, as efficiency is
a critical part of
algorithms. Since you were
working with matrices, discuss how you dealt with the issue of static arrays
(fixed size) vs dynamic data structures, since each
has its pros and cons from a programmers point of
view. In the final analysis, what we are
looking for here is what experience you have gained relative to your
development as a computer scientist.
To all majors: obviously
this is not something you could answer on the spur of the moment on Thursday.
On the other hand, given 4 + days to think about it, you should be able to come
up with some reasonable work.
More fundamentally the
question we are asking here is: mathematically, what in this course is relevant
to my major? This, coupled with the project and team experience you have
gotten, then defines the value of this course beyond merely acquiring 1/3 unit.
Please let me know if
you have questions; I will be happy to talk about any of this with anyone! Please talk to people in your project team.