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Scientia et Technica Año XXVIII, Vol. 28, No. 04, octubre-diciembre de 2023. Universidad Tecnológica de Pereira. ISSN 0122-1701 y ISSN-e: 2344-7214

Linear Algebra Concepts with SageMath for

Systems Engineering students

Conceptos del Álgebra lineal con SageMath para estudiantes de Ingeniería de

Sistemas

S. I. Ceron-Bravo ; R. M. Romero-Luna

DOI: https://doi.org/10.22517/23447214.25171

Scientific and technological research paper

Abstract— This work presents an innovative proposal for the

teaching and learning of systems of linear equations, graphical

interpretation in the plane, in space and orthogonal projection, in

the subject of Linear Algebra, mediated by the SageMath software

as a technology of learning and knowledge. The proposal was

carried out in some Linear Algebra courses in the Systems

Engineering career at the University of Cauca, helping to increase

its quality, reduce the retention rate and contribute to educational

innovation in the university student population. The methodology

for the development of this work consisted of the design and

application of activities that comprise a set of problems which

require the creation of functions in the Python programming

language under the SageMath, environment to obtain possible

solutions and correlate algebraic and geometric representation

registers of the mathematical objects necessary for its solution. We

conclude at the end of this educational research the relevance of

the integration of technological tools within the classroom in

conjunction with active methodological strategies to stimulate the

understanding of some concepts of Linear Algebra; since the

student can visualize, manipulate and observe these abstract

mathematical objects, eliminating unnecessary manual actions

and focusing on the analysis of logical deduction for the solution of

the proposed activities.

Index Terms— ICT; educational innovations, algebra;

geometry; equations

Resumen— Este trabajo presenta una propuesta innovadora para

la enseñanza y aprendizaje de sistemas de ecuaciones lineales, su

interpretación gráfica en el plano, en el espacio y representación

de la proyección ortogonal, en la asignatura de Álgebra Lineal,

mediados por el software SageMath como una tecnología del

aprendizaje y el conocimiento. Esta propuesta se llevó a cabo en

algunos cursos de Álgebra Lineal en la carrera de Ingeniería de

Sistemas de la Universidad del Cauca, contribuyendo a

incrementar la calidad de este, disminuir el índice de retención y

aportar a la innovación educativa en la población estudiantil

universitaria. La metodología para el desarrollo de este trabajo

consistió en el diseño y aplicación de actividades que comprenden

un conjunto de problemas los cuales, en algunos casos, requieren

la creación de funciones en el lenguaje de programación Python

bajo el ambiente de SageMath, para obtener posibles soluciones y

correlacionar registros de representación algebraico y geométrico

de los objetos matemáticos necesarios para su solución.

This manuscript was submitted on September 19, 2022, accepted on September

12, 2023 and published on December 15, 2023.

Concluimos al término de esta investigación educativa la

pertinencia de la integración de herramientas tecnológicas dentro

del aula de clase en conjunción con las estrategias metodológicas

activas para estimular la compresión de algunos conceptos del

Álgebra Lineal; ya que el estudiante puede visualizar, manipular

y observar estos objetos matemáticos abstractos eliminando

acciones manuales innecesarias y centrándose en el análisis de la

deducción lógica para la solución de las actividades planteadas.

Palabras claves—

Álgebra, ecuación, geometría,

innovación

educativa, TIC.

INTRODUCTION

n the academic programme of Systems Engineering at the

Universidad del Cauca, we find as a requirement the subject

Linear Algebra in the second semester (Unicauca, 2022). This

initiative aims to respond to the curricular reflection in the

Faculty of Electronics and Telecommunications Engineering at

the University of Cauca, which aims to improve the teaching

and learning processes of undergraduate programmes in order

to increase their quality, reduce retention rates and contribute to

innovation in university education, as stated by (Maya & Pino,

2019).

The high retention rates in this subject have been evidenced in

other universities, motivating studies that determine this low

performance as shown in the article (Arias, Manco, & Uzuriaga,

2010); likewise, strategies have been proposed to solve

difficulties due to the formalism of linear algebra through

geometry as in (Dorier, Robert, Robinet, & Rogalsiu, 2000); or

on the other hand, due to the teaching experience, also, deficient

previous knowledge of the students in the subjects that demand

the handling of some topics of Linear Algebra for its optimal

development is observed, due to the level of complexity in the

formalism and the connection between algebraic and geometric

thinking in the teaching and learning of this as it is evidenced

in: (Oktac, Sierpinska, & Anadozie, 2002) involving theoretical

thinking, (Ortega, 2002) using computer systems of algebraic

calculation, (Oktac & Trigueros, 2010) involving APOE theory,

(Betancourt, 2014) including digital technologies, (Martinez &

Vanegas, 2021) presenting a didactic sequence based on Van

Hiele's model.

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The aforementioned motivated to give a proposal to develop

educational innovation actions as proposed by (Leal, Rojas,

Ortiz, & Monrroy, 2020) which is achieved through the

transformation of academic activities and not only the

incorporation of new resources so "innovating means opening

horizons, generating an investigative interest, enjoying the

pleasure of inquiring, discovering, proposing, evaluating, but

above all of inventing. It is about advancing in a critical stance

towards existing postulates, because only when it becomes a

subject of reflection, research and questioning is it possible to

innovate" (Salcedo, 2016, as cited in Leal, Rojas, Ortiz, &

Monrroy, 2020) on the other hand "despite the ethereal nature

of the term, pedagogical innovation alludes to the

systematisation and recognition of transformative practices and

as such is an opportunity to make visible and recognise

pedagogical practices according to the needs of the context"

(Gómez 2016, as cited in Leal, Rojas, Ortiz, & Monrroy, 2020).

This is how the incorporation of the mathematical software

SageMath (SageMath, 2022) is proposed, of which we have the

manual (Bard, 2014) and for the specific application of this

software to Linear Algebra with the book (Hefferon, 2021).

We consider this SageMath software as a TAC (Technology of

Learning and Knowledge) since we intend to guide and focus

the Information and Communication Technology (ICT) for

educational uses together with an active methodology in order

to improve the teaching-learning process in the classroom as

suggested by (Unir, 2021), (Lozano, 2011), (Moya, 2013),

corresponding to the need raised in the conclusions of the

conference (Maya & Pino, 2019) to innovate in Engineering

education and attending to multidisciplinarity from the

teaching-learning of mathematics and in particular for the study

of some topics of the subject Linear Algebra, such as solutions

of systems of linear equations and geometric interpretation in

the plane and in three-dimensional space, given the difficulties

of students to face the study of these topics as stated in

(Rodriguez, 2011), (Avilez, Romero, & Vergara, 2016),

(Rosales, 2010).

SageMath is a free program, which can be used as an online or

desktop application, and also allows working with the Python

programming language, which is used in some of the activities

of this project at a basic level, to create small routines and

provide solutions to application problems where conceptual

knowledge of Linear Algebra is required, thus managing to

incorporate them in a transversal way to programming skills

that this academic programme promotes as considered by

(Bravo, Cedeño, Coello, Coello, & Guerrero, 2019).

This proposal seeks to close the gap between the way current

students learn and the traditionally implemented methodology,

highlighting the importance of the implementation of

technological tools in the study of linear algebra and how they

contribute to an interactive didactic between teachers and

students, obtaining results such as: low retention in the course

which is evident in the results obtained, better use of teacher-

student meetings, as well as greater receptivity on the part of

the latter towards the subject.

To implement this idea, a guide was created in pdf format

containing the topics corresponding to the subject of Linear

Algebra with applications focused on the Systems Engineering

programme. This guide, as the topics are developed, presents

instructions on how to use SageMath by means of concrete

examples.

Initially, the software is used to perform numerical calculations

and simplify problem solving. As the course progresses,

routines are implemented through functions created in Python,

under the SageMath environment, which challenge the student

to apply their basic programming knowledge and create

routines to solve problems in Linear Algebra. Engaging the

student in a learning environment characterised by problem

analysis, exploration, discovery, conjecture and verification of

results.

The designed guide includes workshops to enhance the

student's ability to analyse, demonstrate, verify, interpret,

conjecture and apply the different concepts of linear algebra.

This paper presents the results of this didactic research, in the

teaching and learning of some concepts of Linear Algebra, at

the end statistical data are presented that allow us to identify the

impact on the academic performance of students when the

subject of Linear Algebra is developed with the support of

SageMath in the academic periods from 2014 to 2022.

II.

METHODOLOGY

For the development of this work, qualitative research is chosen

as a method, since, through it, a problem is studied that allows

the collection and analysis of direct data of reality in this case

of the solutions proposed by the students through the Sage-

Math software, and activities that allow to observe more

frequent errors in the resolution of the proposed problems. Then

an interpretation is made with the students' notes in the course

of six semesters.

According to Belloch, the stages of evolution in the use of

technological resources by teachers are: access, adoption,

adaptation, appropriation and invention (Belloch, 2012), this

work has been developed in an evolutionary way in the first

three stages, since it was necessary to train in the use of the

mathematical software SageMath, identify activities with this

software for the punctual support in the subject of Linear

Algebra aimed at systems engineering students and finally the

technology has been integrated with the previous knowledge of

programming that students have for the teaching-learning

process of this subject.

In methodological terms, the students are guided by the teacher

in the study of topics related to the solution of systems of linear

equations and proposed problems that require the creation of

functions in the Python programming language under the

SageMath environment to obtain possible solutions. For this

study, a problem is identified as an activity that requires

different complex cognitive processes to work together to find

the mechanism that allows them to be solved, mediated by the

technological-computer resource: SageMath software that

enhances students' ability to discern the correct information and

generate the environment to solve the problem, which cannot

be solved automatically and requires relating diverse

knowledge, on the other hand they do not necessarily have a

single solution path, allowing them to generate and modify

knowledge while developing skills and abilities by practising

what they have learnt, awakening their interest and motivation.

This work seeks to provide challenges and strategies in the

teaching of Linear Algebra, which, as in (Andrews, Berman,

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Fig. 1. Geometry of the line in ℝ

. Source: self-made.

Stewart, & Zandieh, 2018), illuminate particular theoretical

developments on the learning of this science and promise to

engage students to promote their understanding.

The didactic proposal that has been implemented in this study

includes activities formulated based on the texts: (Grossman,

2008), (Martinez & Sanabria, 2014), (Kolman & Hill, 2006),

some of which are presented below. It should be noted that the

student can work from their mobile phone, pc or tablet, with the

SageMath software online or desktop.

The general objective of these activities is to stimulate learning

the solution of systems of linear equations, their geometric

interpretation in cases where the number of unknowns is less

than or equal to 3, in applications to problems in different areas

mediated by SageMath and identifying behaviours of different

mathematical phenomena to analyse, classify, deduce,

conjecture, characteristics and properties.

III.

ACTIVITIES

Activity 1. Lines and Planes in

ℝ

and

ℝ

Aim of the activity:



To develop students' analytical and abstract thinking

skills by asking them for a strategy to solve a situation

using SageMath.



Motivate students by giving them the possibility of

implementing the knowledge of the Systems

Engineering programme.



Stimulate the intuitive part of the concept of a straight

line in ℝ

and ℝ

Define a function in SageMath to find the equation of

the line in ℝ

and another one to print its graph, given

two points in the plane and using determinants theory.

Finally, use an example to run the function.

Solution:

ii.

Create a function in SageMath that calculates the

parametric equation of the line in R3 and another that

prints its graph, if two points through which it passes

are known. Then call the function to find the equation

of the line passing through the points 𝑃

(2, 3, −4) and

𝑃

(3, −2, 5).

Solution:

Fig. 2. Equation of the line in ℝ

. Source: self-made

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C. Activity 2 Ortogonal Proyection

Aim of the activity:



To stimulate the intuitive idea of the concept of

orthogonal projection in the student.



To develop the student's level of deduction in order to

order and direct their ideas.



Encourage the use of the student's basic knowledge of

programming to visualise the orthogonal projection.

i. If 𝑢̅ ≠ 0 and 𝑣̅ ∈ ℝ

𝑛

, we define 𝑝𝑟𝑜𝑦

𝑢̅

𝑣̅ the

orthogonal projection of the vector 𝑣̅ onto 𝑢, as the

vector

𝑣

𝑝𝑟𝑜𝑦

𝑢̅

𝑣̅ =

(

‖‖

)

𝑢̅

Fig. 3. Geometry of the line in ℝ

. Source: self-made

iii.

Create a function using SageMath to find the equation

of the plane containing three non-collinear points. Use

this function to determine the equation of the plane

passing through the points (2, −1,2), (−1,0,3) and

(4, − 3, 4).

Solution:

Conclusions of the activity:

Routines were obtained where it is necessary to identify the

components that define the equation of a straight line and a

plane. This allows a deep understanding and appropriation of

these concepts.

Define a function in SageMath that calculates the

projection of one vector onto another and prints the

graph of , 𝑣̅ and 𝑝𝑟𝑜𝑦

𝑢̅

𝑣̅. Show with an example its

use.

Use the function created above on two fixed

vectors 𝑢̅ , 𝑣̅ of ℝ

and calculate the projection u, v.

Show with an example its use.

By calling 𝑣

𝑐

= 𝑣̅ − 𝑝𝑟𝑜𝑦

𝑢̅

what conclusions can we

infer if

𝑢̅ and 𝑣̅ are orthogonal

II.

𝑢̅ y 𝑣̅ and v are parallel

III.

Create a function that reports by means of a

message what happens in each case and prints

the graphs.

Solution a:

Fig. 5. Orthogonal projection. Source: self-made.

Solution b.

Fig. 6. Orthogonal projection calculation. Source: self-made.

Solution 1c: To give this solution the student relied on the dot

product and cross product to determine whether the vectors

were parallel or orthogonal.

Fig. 4. Plane equation in ℝ

. Source: self-made.

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Fig. 7. Function orthogonal projection. Source: self-made.

Fig. 8. Geometry of the orthogonal projection when the vectors are

orthogonal using Fig. 7. Source: self-made.

Fig. 9. Geometry of the orthogonal projection when the vectors are

parallel using Fig. 7. Source: self-made.

Solution 2c: In this solution the student calculates the angle

between two vectors to determine whether the vectors are

parallel or orthogonal.

Fig. 10. Function orthogonal projection using dot product and cross

product. Source: self-made.

Fig. 11. Geometry of the orthogonal projection when the vectors are

orthogonal using Fig. 10. Source: self-made.

Fig. 12. Geometry of the orthogonal projection when the vectors

are parallel using Fig. 10. Source: self-made.

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Fig. 13. Solution of homogeneous linear systems of equations that have the

same number of equations as unknowns using the range of matrices.

Source: self-made.

Fig. 15. Solution of homogeneous linear systems of equations that have the

same number of equations as unknowns, using the inverse matrix theory.

Source: self-made.

Conclusions of the activity:

The power of visualisation that SageMath allows makes the

student obtain an intuitive idea of the orthogonal projection of

one vector on another and establish criteria between u and v that

allow him to infer relationships between the projection vectors

v and vc. It is noteworthy that having different solutions

enriches the feedback experience, exploring the different results

of Linear Algebra.

Activity 3. Solving systems of linear equations

Aim of the activity:



To guide the student to explore and implement the

results of determinant, inverse matrix and the

application these have to the solution of non-

homogeneous systems of linear equations.



Motivate students to develop ideas from their area of

study to provide solutions to problems that are

modelled by means of systems of linear equations.

Create a function in SageMath that receives as parameters

the matrix of coefficients and the vector of independent

terms of a system of non-homogeneous linear equations

with number of equations equal to the variables, the

function should be as efficient as possible to report:



If the system has only one solution, it must output

a message and calculate the solution.



In case the system has infinite solutions, it must

give a message informing and calculating the

solution set.



The function must contemplate the case in which

the system has no solution and report it.

Solution 1: In this solution the student uses matrix rank theory

to create the function.

Solution 2: This solution presents an error, it takes advantage of

the student's error where he does not take into account what

could happen if a system despite having a determinant equal to

zero does not have a solution. Here, in the example, although

the function is in error, SageMath does indicate that the system

has no solution by means of a message. This example is used to

confront the student's analysis.

Fig. 14. Solution of homogeneous linear systems of equations that have

the same number of equations as unknowns, making use of the

determinant theory. Source: self-made.

Solution 3: In this case the student uses theoretical aspects of

the inverse of a matrix to calculate the solution in the case of

having only one solution, but does not analyse the case of

infinite solutions, nor does he calculate the solution set and the

one with no solution. The execution of the function is shown

for a particular system.

Solution 4: Here we see another solution, the student succeeds

in calculating the solution set when the system has a coefficient

matrix of size three by three. It does indeed work, but only if

the system has at least one solution. The execution of the

function to a particular system is shown.

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Fig. 16. Solution of homogeneous linear systems of equations that have the

same number of equations as unknowns, making use of the determinant

theory. Find the solution for the case where the system has three unknowns

and three equations. Source: self-made.

Fig. 17. Counter example. The error that can be presented by the functions

created by the students to solve systems of linear equations is indicated.

Source: self made.

Fig. 18. Application of systems of linear equations. Source: self-made

In case the system has no solution, the function shows a

contradiction, which is used in class session for feedback with

the following example: the function is called and evaluated on

a zero matrix associated to a non-homogeneous system whose

determinant is equal to zero and that the system has no solution,

the function although calculating the solution set correctly the

message "the system has infinite solutions" contradicts that

solution.

These solution types are analysed in the classroom in order to

detail errors in the solution of the activity by means of feedback

and to inform on how to overcome these difficulties by means

of a feedforward.

2. Use the function created to solve the following problem and

interpret the solution: a state fish and game department provides

three types of food to a lake that is home to three species of fish.

Each fish of species 1 consumes an average of 1 unit of food 1,

1 unit of food 2 and 2 units of food 3 each week. Each fish of

species 2 consumes an average of 3 units of food 1, 4 units of

food 2 and 5 units of food 3 each week. For a fish of species 3,

the average weekly consumption is 2 units of food 1, 1 unit of

food 2 and 5 units of food 3. 25,000 units of food 1, 20,000 units

of food 2 and 55,000 units of food 3 are provided to the lake

each week. If we assume that the fish eat all the food, how many

fish of each species can coexist in the lake?

Using the function created in solution 1 of item 1, we obtain:

For fish to coexist it is concluded that 5000<z<8000, since the

number of fish of each species must be strictly greater than zero.

Conclusions of the activity:

There are students who manage to interpret according to the

data of the problem the solution that the software yields by

means of the created function. Others do not realise the domain

of the free variable, which in this case is z, so that the fish could

coexist.

Activity 4. Geometric interpretation of the solution of

systems of linear equations in ℝ

Aim of the activity:



To implement SageMath to graph the geometric objects

corresponding to the equations of a system of linear

equations with three unknowns and its solution, so that

from observation the student obtains an intuitive idea of the

type of solution set.



Establish a relationship between the solution set of systems

of linear equations and the intercept of planes in space.



Coordinate the algebraic and graphic representation

registers in the process of solving systems of linear

equations.

Consider the following system of linear equations.

𝑥 + 2𝑦 − 𝑧 = 0

𝑥 + 2𝑦 − 𝑧 +

= 0

Use SageMath to graph the equations of the system and to infer

the type of solution set

Solution:

Fig. 19. Geometry of the solution set type of a system of linear

equations with two equations and three unknowns, when the solution

set is empty. Source: self-made.

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Fig. 20. Geometry of the solution set type of a system of linear equations

with three equations and three unknowns, when the solution set is infinite.

Source: self-made.

Fig. 22. Equation and geometry of the solution of systems of linear

equations with three equations and three unknowns, when the

solution set is a point. Source: self-made.

Fig. 23. Equation and geometry of the solution of systems of linear

equations with three equations and three unknowns, when the solution set

is empty. Source: self-made.

ii.

Consider the following system of linear equations.

2𝑥 + 3𝑦 − 4𝑧 = 2

4𝑥 + 6𝑦 − 8𝑧

= 4

6𝑥 + 9𝑦 − 12𝑧

= 6

Use SageMath to graph the equations of the system and infer

the type of solution set.

Solution:

iii.

Consider the following system of linear equations.

2𝑥 + 3𝑦 − 4𝑧 + 5 = 0

3� + 2� + 5� + 6 = 0

Solve the system in SageMath. Interpret geometrically each

equation and the solution set of the system.

Fig. 21. Equation and geometry of the solution of systems of linear

equations with two equations and three unknowns, when the solution set

is infinite. Source: self-made.

iv.

Give three systems of three-by-three equations with

different solution sets and relate the type of solution set to

the graph.

Solution:

Three planes whose intersection is a point.

Three parallel planes

Three planes that do not intersect at the same time

Fig. 24. Equation and geometry of the solution of systems of linear

equations with three equations and three unknowns, when the solution set

is empty. Source: self-made.

Three planes that do not intersect at the same time.

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Fig. 25. Equation and geometry of the solution of systems of linear

equations with three equations and three unknowns, when the solution set

is infinite. Source: self-made.

Conclusions of the activity:

The visualisation allowed by SageMath makes it possible for

the student to obtain a geometric interpretation of the type of

the solution set of the system. By correlating the graphical

representation of the solution set with its algebraic

representation, the student can identify the need to modify the

domains of the variables and thus fully understand the meaning

of the solution set.

Activity 5. Geometric interpretation of linear

independence/dependence between vectors of ℝ

Aim of the activity:



To study the different tools of SageMath together with the

theoretical results of Linear Algebra for the study of linear

dependence and independence between vectors of the usual

vector space ℝ

Consider the following definition and solve:

Let

{

�

, �

, . . . , �

�

}

be a set of vectors in a vector space

V. These vectors are linearly dependent (LD) if there exist

� scalars �

, �

, . . . , �

� ,

not all null such that:

�

+ �

�

+ �

�

+. . . +�

�

= 0.

If the vectors are not linearly dependent they are said to be

linearly independent (LI).

Fig. 26. Analysis of linear dependence/independence of vectors in ℝ

Using the coefficient matrix scaling of the associated homogeneous system

resulting from posing the solution to the problem. Source: self-made.

Solution 2: As the matrix is square, the student uses the

functions created in activity three, given that to answer this

question it is necessary to know the solution set of the system

of homogeneous linear equations since we are considering the

trivial zero in the usual vector space ℝ

. The student uses the

fact that if the system has only one solution it must be the trivial

one, and in this case the vectors are LI, if the system has

solutions other than the trivial one the vectors are LD.

Fig. 27. Analysis of linear dependence/independence of vectors in ℝ

Using determinant theory applied to the coefficient matrix of the associated

homogeneous system resulting from posing the solution to the problem.

Source: self-made

Another way in which SageMath can help answer whether a set

of vectors is LI or LD, in a more simplistic way which does not

allow the student to explore the necessary theory but facilitates

the creation of functions to make the graphs is: with the

command “are_linearly_dependent(vecs)” For example:

1 0 −1

Determine whether the set of vectors {[

] , [

−2

]}

0 1 3

are LI or LD, in the usual vector space ℝ

Solution 1: In this solution the vectors are written as columns

of a matrix that is associated to a homogeneous system, the

matrix is scaled, the student deduces from the scaled form of

the matrix whether the vectors are LD or LI, using the

definition.

Fig. 28. Analysis of linear dependence/independence of vectors in ℝ

Using SageMath's own functions. Source: self-made.

Create a function that receives in its argument three

vectors of the usual vector space R3 and returns if the set

of these three vectors are LI or LD and the graph.

Illustrate its use with examples.

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Fig. 29. Geometry of the linear independence of vectors in ℝ

. Source:

self-made.

Fig. 30. Geometry of the linear dependence of vectors in ℝ

. Source: self-

made.

Fig. 31. Geometry of the linear dependence of vectors in ℝ

. Three vectors

in the same plane. Source: self-made.

Solution:

We see in the student's solution the use of SageMath's own

commands for the creation of a new function, which in this case

indicates whether the vectors are LI or LD, but in addition

returns the graph of the vectors.

Example 1:

Example 2:

Example 3:

Conclusions of the activity:

Item 1 of this activity makes it possible to relate the definition

of linearly dependent and independent set to the solution of a

system of linear equations its associated scalar matrix, on the

contrary, item 2 provides an automatic solution of this

definition, but with a visual potential that allows to identify that

a set of three vectors in the usual vector space ℝ

are linearly

independent if they lie in different planes.

IV.

RESULT

An analysis of the courses AL-2014 group D, AL-2014 group

K, AL-2016 group J, AL-2018 group J, AL-2018-2, AL-2019

group A, AL-2019 group Q, AL-2019 group J, AL-2020 group-

I, AL-2021 group F and AL-2021 group Q, of the subject Linear

Algebra with students of the systems engineering programme

of the Universidad del Cauca is presented below, using

frequency histograms and infograms that allow inferring the

influence of the use of the SageMath software in the retention

of these courses. The histograms that show the frequency of the

final grades obtained by the students in each Linear Algebra

course in the different years and academic periods mentioned

are described below.

The following histograms are the one corresponding to the

course AL-2014 group D. In this course 75% of the students

had grades less than or equal to 2.85 and only 26% passed and

other histogram of the course AL-2014, group K (Linear

Algebra group K of the year 2014). In this course 50% of the

students had final grades less than or equal to 2.5 and 49%

passed the course.

Fig. 32. Final grades corresponding to group J and group K of

Linear Algebra in the year 2014 respectively. Source: self-made.

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Fig. 36. Final grades corresponding to group I 2020, group F and Q

of Linear respectively Algebra in the year 2021. Source: self-made.

We continue with the histograms corresponding to the final

grades of the courses AL-2016 and the course AL-2018 group

J, in the first case 50% of the students have grades below 2.5

and only 26% pass the course and in the second case the 50%

of the students had final grades less than or equal to 2.2 and

only 38% pass the course.

Fig. 33. Final grades corresponding to groups J of Linear

Algebra in the years 2016 and 2018. Source: self-made.

For the year 2018 in the second period, the SageMath software

is incorporated with the methodology presented in this work, in

the course AL-2018, 95% of the students passed the course,

beginning to show a low retention in the subject.

Likewise, in the AL-2019 group A course, 67% of the students

pass the course, a slightly lower percentage, but still a good

percentage of students who pass compared to the courses in

which SageMath was not used.

Fig. 34. Final grades corresponding to groups 2 and group A of Linear

Algebra in the years 2018 and 2019 respectively. Source: self-made.

In the courses AL-2019 group J 64% of the students passed and

group AL-2019 group Q the 81% of the students passed the

course.

Fig. 35. Final grades corresponding to group J and group Q of

Linear Algebra in the year 2019 respectively. Source: self-made.

Finally, in the years 2020 and 2021 it is notherworthy that we

were in virtuality due to the pandemic, circumstance that limits

the work given the difficulty of access of the students to

computer equipment and internet, this caused a premature

withdrawal by the students.

Although there was a decrease in the number of students who

passed the subject Al-2020 group I, it is still a good result. Here

50% of the students passed. In the AL-2021 group F course, we

see a high number of students passed the course, to be more

precise, 81%. In the course of Al 2021, group Q, the percentage

of students who pass the course drops to 50%. It is worth noting

that, when implementing an innovative pedagogical strategy,

not all courses respond in the same way, but even so, the results

obtained show a drop in retention.

To illustrate the impact on student retention in this subject, the

following graph shows the courses in their respective periods

versus the percentage of students who passed. In the period

between 2014 and 2016, we recall, the courses were developed

through lectures, tutorials, written mid-term exams, while

between 2018 and 2021 the subject is developed with the

support of SageMath, and the methodology already explained.

Thus, the percentage of students who pass the subject before

incorporating this methodological strategy and after

implementing it is compared.

Fig. 37. Each point indicates the percentage of students

passing before incorporating SageMath with activities

CONCLUSIONS

It is well known that the assimilation and integration of the

concepts inherent to Linear Algebra represent a challenge for

students given the level of abstraction of some of the topics

studied in this subject. This proposal undoubtedly shows that

the support of technological tools intercepted with a

methodological strategy makes a great difference in the

understanding

and

appropriation

the

concepts

that

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Scientia et Technica Año XXVIII, Vol. 28, No. 04, octubre-diciembre de 2023. Universidad Tecnológica de Pereira

reflected in the academic results according to the final grades

of each student.

Innovation in the educational field focused on the use of ICT as

a tool for training purposes TAC, allows particularly in the

study of Linear Algebra to contribute to improving the training

of engineers for the future, responding to the need for curricular

changes and teaching-learning processes that reduce desertion

in this subject, ensure the quality of learning and contribute to

social cohesion with the same interests and common goals

among the student population and teaching staff.

Thanks to the reflection of the teaching work and the interaction

with the student in the development of the chairs, these

innovative ideas arise that leave a reference and invite other

teachers to explore new methodologies that represent a change

in the way of teaching and learning mathematical concepts. In

this spirit, we intend to continue integrating and combining

various technologies in the exploration of knowledge and

analysis of algebraic objects.

ACKNOWLEDGMENTS

The authors thank the Universidad del Cauca for granting the

respective endorsement to the internal development research

project with ID 5824, entitled " TIC y TAC aplicadas al Álgebra

Lineal ", within the group " Estructuras Algebraicas,

Divulgación Matemática y Teorías Asociadas "

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