Final report applied calculus for it

INTRODUCTION AND OVERVIEW 1.1 The reason for choosing the topic Linear algebra is a branch of mathematics that studies vector spaces,systems of linear equations, and linear transformatio

TON DUC THANG UNIVERSITY

FACULTY OF INFORMATION TECHNOLOGY

NGÔ QUỐC VINH - 52300085

PHAN ANH KHOA - 32001031

FINAL REPORT

APPLIED CALCULUS FOR IT

HO CHI MINH CITY, 2023

VIETNAM GENERAL CONFEDERATION OF LABOR

TON DUC THANG UNIVERSITY

PHAN ANH KHOA - 32001031

FINAL REPORT

APPLIED CALCULUS FOR IT

Advised by

Dr Vo Tran An

We would like to express our deep gratitude to Professor Vo Tran An for his invaluable guidance and unwavering support throughout the entire process of completing this report Professor Vo Tran An 's dedication to our academic growth, coupled with his wealth of knowledge and expertise, has been instrumental in shaping our understanding and approach to the subject matter He consistently went above and beyond to provide insightful feedback, encouragement, and direction, which greatly enriched our learning experience.

We are deeply appreciative of the time and effort that Professor Vo Tran An invested in mentoring us, patiently addressing our inquiries, and challenging us to strive for excellence His mentorship has not only enhanced our academic capabilities but also instilled in us a sense of confidence and determination to tackle future challenges Furthermore, we are grateful for Professor Vo Tran An 's willingness to share his expertise and insights, which have undoubtedly broadened our perspectives and inspired us to explore new avenues of research and inquiry.

In closing, we would like to express our sincere appreciation for the profound impact of Professor Vo Tran An on our academic journey His guidance has been truly transformative, and we feel fortunate to have had the opportunity to learn under his mentorship.

With deepest gratitude,

Ho Chi Minh city, 3 September 2023 rd

Author

Phan Anh Khoa Ngo Quoc Vinh

DECLARATION OF AUTHORSHIP

We hereby declare that this is our own project and is guided by Mr Vo TranAn; The content research and results contained herein are central and have not beenpublished in any form before The data in the tables for analysis, comments andevaluation are collected by the main author from different sources, which are clearlystated in the reference section

In addition, the project also uses some comments, assessments as well asdata of other authors, other organizations with citations and annotated sources

If something wrong happens, we will take full responsibility for the content of my project Ton Duc Thang University is not related to the infringing rights, the copyrights that We give during the implementation process (if any)

Ho Chi Minh city, 3 September 2023 rd

Author (Signature and full name)

Phan Anh Khoa Ngo Quoc Vinh

TABLE

LIST OF FIGURES

LIST OF TABLES

ABBREVIATIONS

CHAPTER 1 INTRODUCTION AND OVERVIEW

1.1 The reason for choosing the topic

1.2 Project Objectives

CHAPTER 2 THEORETICAL BASIS

2.1 Linear Algebra

2.1.1 History

2.1.2 Linear Algebra Applications

2.2 The theorems and properties being applied

CHAPTER 3 DESIGNING AND IMPLEMENTING

3.1 Some problems

3.2 Solve problem

3.2.1 Given the matrix A Find all values of a for which det(A) = 0

3.2.2 Solve the following system of linear equations

3.2.3 Let v = (1 ; 1; 1), v = (2; 5; 1), v = (3; 0; 5) 1 2 3 3.2.4 Find a matrix P that diagonalizes

3.2.5 Let S = {v =(2;4;3), v = (2;4;2), v = (-6;4;2)} 1 2 3 3.2.6 Use the Gram-Schmidt orthonormalization process to transform

CHAPTER 4 RESULT

4.1 Conclusion

4.2 Potential and Development Direction

REFERENCES

LIST OF FIGURES

LIST OF TABLES

PROJ Projection

CHAPTER 1 INTRODUCTION AND OVERVIEW

1.1 The reason for choosing the topic

Linear algebra is a branch of mathematics that studies vector spaces,systems of linear equations, and linear transformations between them It plays afundamental role in the development of mathematics and serves as a cornerstone inmodern higher−level mathematical curricula Linear algebra finds extensiveapplications in various mathematical fields such as abstract algebra, functionalanalysis, and analytic geometry Moreover, it has countless applications in naturalsciences (physics, engineering, etc.) and social sciences Determinants are crucialtools in linear algebra and have numerous applications in mathematics Themethod of determinants provides a concise and clear approach to mathematicalconcepts and serves as an effective problem−solving method It positivelycontributes to the development of logical thinking for learners of mathematics

Linear algebra is a foundational area of mathematics: It serves as thebasis for many other fields such as geometry, calculus, probability theory, andmore

Widespread applications: Linear algebra is used in various industries,including engineering, computer science, economics, physics, and many others

Addressing real−world problems: Linear algebra helps solvenumerous real−world problems such as optimization, data analysis, signalprocessing, and more

Developing logical thinking and analysis: Studying linear algebrafosters the development of logical thinking and analytical skills, as well asproblem−solving abilities.

Career opportunities: There is a high demand for individuals withknowledge of linear algebra in many industries, particularly in technology anddata−related fields

1.2 Project Objectives

Understand fundamental concepts: Master basic concepts of linearalgebra such as matrices, vectors, vector spaces, eigenvalues, eigenvectors, andrelated operations

Develop problem−solving skills: Enhance the ability to apply thetheory and techniques of linear algebra to solve real−world problems in fields such

as engineering, computer science, physics, and economics

Apply to modern technology: Understand how linear algebra isapplied in modern technologies such as artificial intelligence, machine learning,image processing, and neural networks

Research and development: Conduct research to expand knowledge oflinear algebra and develop new methods or improve existing ones

Prepare for careers: Equip students with the necessary knowledge andskills to pursue careers in fields related to linear algebra

Foster critical thinking: Strengthen analytical, evaluative, and criticalthinking skills through solving problems and challenges related to linear algebra

Academic exchange: Create opportunities for students and researchers

to exchange knowledge and experiences with the academic community and relatedindustries

CHAPTER 2 THEORETICAL BASIS

2.1 Linear Algebra

2.1.1 History

Linear algebra is a branch of mathematics that evolved from the study of linearequations and systems of linear equations Its history can be traced back throughseveral centuries, with significant contributions from various mathematicians.Ancient Civilizations: Ancient civilizations such as Ancient Egypt and Babylonemployed linear methods to solve problems related to measurement andcommerce

Ancient Greece: The Greek mathematician Euclid introduced fundamentalconcepts of geometry, which later formed the basis for linear algebra

Renaissance Period: Italian mathematician Gerolamo Cardano solved systems oflinear equations in his studies of algebra in the 16th century

17th and 18th Centuries: German mathematician and astronomer Johann CarlFriedrich Gauss developed the method of Gaussian elimination, a significanttechnique in linear algebra for solving systems of linear equations

19th Century: Augustin−Louis Cauchy contributed to the development ofdeterminant theory, and Arthur Cayley laid the groundwork for matrix theory.20th Century: Linear algebra began to emerge as a distinct mathematical field withthe development of vector theory and vector spaces

Advancements in this theory have been propelled by the work of mathematicianssuch as Giuseppe Peano, Hermann Grassmann, and many others

In the modern era: With the development of computers and informationtechnology, linear algebra has become an indispensable tool in various fields such

as computer science, engineering, physics, economics, and many others

Today, linear algebra is an integral part of mathematical education and is widelyapplied in both research and industry

2.1.2 Linear Algebra Applications

Linear algebra has numerous real−world applications, spanning from naturalsciences to engineering and social sciences Below are some specific applications:

Engineering: In electrical and mechanical engineering, linear algebra is used toanalyze electrical circuits, mechanical structures, and dynamic systems

Computer Science: Algorithms related to computer graphics, image processing,artificial intelligence, machine learning, and optimization all utilize linear algebra

Economics and Finance: Linear algebra is used to model and solve optimizationproblems in economics, risk management, and investment analysis

Physics: Physical theories such as quantum mechanics, relativity, andelectromagnetism rely on linear algebra

Statistics and Data Science: Regression analysis, principal component analysis(PCA), and many other statistical methods require linear algebra

Natural Sciences: In chemistry and biology, linear algebra is used to model molecular

Simulation and Modeling: Linear algebra is an essential tool for simulatingphysical and engineering systems, from weather models to fluid flow simulations.

Communication and Signal Processing: Methods such as Fourier transformand Laplace transform, crucial in signal processing, rely on linear algebra

Automatic Control: Linear algebra is used to design and analyze automaticcontrol systems and robots

Neural Networks and Deep Learning: Models in deep learning, includingartificial neural networks, use linear algebra to process and learn from large datasets

Overall, linear algebra is a powerful and versatile mathematical tool withapplications in almost every scientific and engineering field

2.2 The theorems and properties being applied

Cramer's Rule: This theorem provides a method for solving systems

of linear equations by expressing the solution as a ratio of determinants If thedeterminant of the coefficient matrix is nonzero, the system has a unique solution.Cramer's Rule expresses the solution in terms of determinants of related matrices

Properties of Determinants: Determinants have properties such assymmetry, block property (determinant of a block matrix), properties of theinverse matrix (determinant of the inverse is the reciprocal of the determinant), andproperties of the transpose matrix (determinant of a transpose matrix is the same asthe original)

Inverse Matrix Theorem: A square matrix has a unique inverse if andonly if its determinant is nonzero The determinant being zero implies the matrix issingular and does not have an inverse.

Minor Determinant Theorem: The determinant of a submatrixremains unchanged if the rows or columns form a linearly independent set If thesubmatrix forms a basis for the vector space, the determinant remains the same

Matrix Multiplication Determinant Theorem: The determinant of theproduct of two matrices is equal to the product of their determinants, i.e., det(AB)

= det(A) * det(B)

Eigenvalue−Determinant Relationship Theorem: The sum of theeigenvalues of a square matrix is equal to the determinant of the matrix Thisrelationship is fundamental in understanding the properties of eigenvalues anddeterminants

Transpose Matrix Determinant Theorem: The determinant of atranspose matrix is equal to the determinant of the original matrix, i.e., det(A^T) =det(A)

Addition or Removal of Row or Column Theorem: Adding orremoving a row or column from a matrix results in a specific change in thedeterminant of the matrix, which can be calculated using the properties ofdeterminants

Laplace's Theorem: Laplace's theorem, or Laplace expansion,provides a method for calculating the determinant of a matrix by using thedeterminants of smaller submatrices It is a useful tool for computing determinantsefficiently.

CHAPTER 3 DESIGNING AND IMPLEMENTING

3.1 Some problems

So, the solution to the system of equations is a = −1

3.2.2 Solve the following system of linear equations

by using Gaussian Elimination method.

Now, let's perform Gaussian elimination to transform this matrix into itsrow−echelon form.

Step 1: Eliminate the coefficient below the leading coefficient in the first column.Multiply the first row by −3 and add it to the second row, then multiply the firstrow by −5 and add it to the third row

So, the solution to the system of equations is =11/8, =1/8, =−1𝑥 𝑦 𝑧

Solve the following system of linear equations by using Gaussian Elimination method.

¿

Step 2: Subtract the second row from the third row to make the coefficient belowthe leading coefficient in the second column zero

3.2.3 Let v = (1 ; 1; 1), v = (2; 5; 1), v = (3; 0; 5) 1 2 3

Show that the set B = { v 1; v 2 ; v } is a basis of 3 R3

To show that the set B = { v1; v2; v3} is a basis of R3, we need todemonstrate two things:

1.Spanning: Show that any vector in R3 can be written as a linear

where c , c and c are scalars.1 2 3

We need to show that for any 𝑥∈R3

There exist v 1 ; v ; v 2 3such that x = c v + c v + c v 1 1 2 2 3 3

This can be done by solving the equation system:

{c1+2 c2 −3c3 =x1

c1 −2c2+0 c3=x2

c1 +c2+5 c3 =x3Where x1,x2 and x3 are the components of x This system of equations can be solvedfor c1 , c2and c3 ensuring that any vector x can be expressed as a linear combination

of v 1 , v , v 2 3

Linear Independence:

To show that v 1 , v 2 , v 3 are linearly independent, we need to show that the

only solution to the equation:

And row reducing it to its row echelon form If the only solution to this system

is the trivial solution c 1 = c = c = 0 2 3 , then v 1 , v 2 and v 3 are linearly independent

By demonstrating both spanning and linear independence, we can conclude that𝐵={ v1; v2; v3}B={ v1; v2; v3 } is indeed a basis for R3

3.2.4 Find a matrix P that diagonalizes

Expanding this determinant, we get the characteristic polynomial:

By solving this equation, we find the eigenvalues to be:

From the second row, 2x + y = 0, which gives x = ½

So, for,λ=1 one eigenvector is v 1 = (1

0)

For λ=1, we can pick another linearly independent vector, for example,

v =3 (0 1

0)

Now, we form the matrix P with the eigenvectors as its columns:

Find the coordinate vector of v =(54,12,9) relative to S

To solve this problem, we need to find the coordinate vector of v = (54,12,9)relative to the basis S = {v1=(2;4;3), v2=(2;4;2), v3=(-6;4;2)}

To do this, we need to solve the linear equation system x v + x v + x v =1 1 2 2 3 3

v, where v , v , and v are the basis vectors, and v is the vector we are looking for1 2 3

the coordinate vector

Given: