Linear Algebra Project: Computer-Aided Exploration of Key Concepts

This project explores introductory linear algebra concepts at the university level, utilizing computer software to aid in calculations and computer graphics to visualize the concepts. The project covers topics such as polynomial curve fitting, linear transformations involving rotation, reflection, and shear matrices, and the properties of determinants. It emphasizes the use of computer algebra systems and graphing calculators to enhance understanding and make mathematics more engaging. The project includes practical applications, such as analyzing light intensity variations and exploring matrix transformations, providing students with hands-on experience and insights into mathematical research and discovery. Desklib offers a wide range of similar solved assignments and past papers for students.

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Abstract
This particular written project seeks to highlight the introductory linear algebra at the
University. The project utilises the computer software that aids in the reputation of
calculations and also allows for the use of the computer graphics to visualize the concepts.
The key words that have been used in this particular project include Mathematics, computer
projects and Linear Algebra.
Introduction
People naturally have a perfect relationship with the linear algebra at the introductory level.
Student love extends to the abstract vector spaces while I tend to show little interest to the
matrix computations (Woodruff 2014). This can be traced back when student started their A-
level linear algebra course. The topic opened with the vectors 1 followed with the vector 2.
They both had systems of equations at the centre with matrix guiding the entire process.
Towards the end of the week was abstract vector spaces (Talbert 2014).
As the learning continued, they were almost in the mid-way when they were required to
major and one could not just believe it all. All other classmates did not seem to share with the
rest this dream. They all enjoyed the course up to the point we were tackling the vector
spaces. After this point things started becoming tough and they subsequently became
miserable. Students continued with their learning program while using the linear algebra as a
necessary tool though some did not give it much of my attention (Eddelbuettel & Sanderson
2014).
When they reached second year of my study, they were assigned a class of our junior to
teach on the materials that we had previously covered. This was not actually a perfect idea to
many. It was however expected of them to be able to teach a course that they had been taught.
They would have engineering and even the computer science as the majors in their course. In

their mind, they knew they needed a way that they could use to enliven the aspects of the
course that needed computer applications. At that particular time, the university was in the
process of creating a mathematics laboratory that would mainly be used for the calculus
operations.
This particular laboratory would be very useful as a tool to the algebra too (Hubbard &
Hubbard 2015).The note as it is structured describes elementary projects that have been
handled by students outside the class environment. Some have to admit that there was no
prior experience with the groups or clubs of mathematics. A series of the problems were
looked into including:
Discussion
Polynomial Curve setting
Taking a group of n data points
And having the degree as n-1. The graph of this particular equation will have to pass through
the points that have been given. From the graph obtained can be shown that if all the
coordinates of X are district then what is obtained with respect to the result is a very unique
polynomial of degree n-1 or less. The respective values of a up to the highest power can be
obtained. This is achievable by solving the equations such as the one shown below with the n
values.

The computer may not really help in the setting up of the system of the relevant equations. It
is however very useful in solving the equations (Mirsky 2012). After I had practiced with the
three data points, I took a very actual data to work on. Some used the intensity of the output
of light as flashes out from the bulb which checking on its variation with the time it was
allowed to ignite. Other than getting the required polynomial, it was easy to quickly have the
visualised results on the computer and have its graphs and polynomial plotting properly done.
The polynomial was used to estimate the values of the figures and equations such as the Y
value and the X value.
Linear Transformation
This was the second area that was addressed in the project work. In this particular case., there
were some sets of the points that were to be taken through the process of rotation, reflection
and changes using the share matrices. The reflection matrix that was used include the
reflection
to the points within the grid whose vertices include (0,0), (0.1),
(1,1) and finally (1,0).
A more interesting reflection matrix taken as

In this particular case there was plotting of the original points and the other points that results
from the other transformations of the matrix. The questions that were being tackled in this
case were the mapping of the points to other points. It also helped in the determination of
whether the names of the identified transformations were appropriate. It was also to check
whether there is a single matrix that entire represents a process of reflection then followed by
rotation. The project was a straight one with a lot of fun.
When the command keys were pressed, they generated a list of points in the square unit that
allowed for the use of the rotation matrix and also on the display was the original points
together with the expected results.
Determination of the properties of the determinants.
This particular focussed on the studying the properties of the determinants in the class
lecture. In consideration was the function of the determinant for the commonly used n*n
matrix while also studying the use of a two by two matrices determinant prior to attempting
the project itself. Computers were used in the computation of the determinant followed by
their transposes. The next procedure was to look at the three by three matrix, then four-by
four matrices up to the points it was possible to draw a very convincing conclusion. There
was also a problem touching on the determination of the determinant of the sum of the two

matrices. Another repeat was done on the product matrix. Having known the required trend
and skill applications, there was exploration on the matrix of choice and this was equally
interesting. This provided a perfect opportunity to feel on how the mathematical researches
are done and how discoveries of mathematics are done as opposed to the memorising
(Wilkinson & Reinsch 2012).
Conclusion
The presence of the computer algebra system and even the scientific calculators that are today
used in the graphing has brought a significant change in the coursers of the linear algebra.
The graphing allows for the visualization of some concepts and topics. This generally make
mathematics a fun.

References
Wilkinson, J. H., & Reinsch, C. (2012). Handbook for Automatic Computation: Volume II:
Linear Algebra (Vol. 186). Springer Science & Business Media.
Mirsky, L. (2012). An introduction to linear algebra. Courier Corporation.
Hubbard, J. H., & Hubbard, B. B. (2015). Vector calculus, linear algebra, and differential
forms: a unified approach (pp. 818-pages). Matrix Editions.
Woodruff, D. P. (2014). Sketching as a tool for numerical linear algebra. Foundations and
Trends® in Theoretical Computer Science, 10(1–2), 1-157.
Talbert, R. (2014). Inverting the linear algebra classroom. Primus, 24(5), 361-374.
Eddelbuettel, D., & Sanderson, C. (2014). RcppArmadillo: Accelerating R with high-
performance C++ linear algebra. Computational Statistics & Data Analysis, 71, 1054-1063.