Definition
“Linear algebra is a branch of mathematics, but the truth of it is that linear algebra is the mathematics of data. Matrices and vectors are the language of data.
Linear algebra is about linear combinations. That is, using arithmetic on columns of numbers called vectors and arrays of numbers called matrices, to create new columns and arrays of numbers. Linear algebra is the study of lines and planes, vector spaces and mappings that are required for linear transforms.” [1]
Linear algebra lies at the heart of mathematics. It allows us to construct and solve systems of linear equations, unifying otherwise disparate topics (read: functional analysis, physics, regression) through a common set of guiding principles and applicable techniques.
A typical introductory course introduces matrices and vectors, finding the four fundamental subspaces of a matrix (check out my article on that topic here), solving systems of linear equations, and modelling problems from related fields, among many other interesting concepts. In this article, we’ll explore why you should start learning linear algebra today. [3]
Who
Anyone who studies linear algebra without understanding the power it has to solve real-world problems.
What
Linear algebra is central to almost all areas of mathematics. For instance, linear algebra is fundamental in modern presentations of geometry, including for defining basic objects such as lines, planes and rotations. Also, functional analysis, a branch of mathematical analysis, may be viewed as the application of linear algebra to spaces of functions.
Linear algebra is also used in most sciences and fields of engineering, because it allows modeling many natural phenomena, and computing efficiently with such models. For nonlinear systems, which cannot be modeled with linear algebra, it is often used for dealing with first-order approximations, using the fact that the differential of a multivariate function at a point is the linear map that best approximates the function near that point. [4]
Why
“The benefits of understanding and learning linear algebra are mentioned in a few of its applications. They are:
- Traffic flow
- Electric currents
- Cryptography
- Graph theory
- Image processing [5]
- Least square approximation
Barring that, linear algebra is used in all forms of engineering applications. It is an essential aspect when it comes to machine learning. Computer-intensive tasks require knowledge of linear algebra.” [2]
See Theoretical Knowledge Vs Practical Application.
How
Many of the References and Additional Reading websites and Videos will assist you with using and understanding linear algebra.
As some professors say: “It is intuitively obvious to even the most casual observer.“
References
[1] Brownlee, Jason. 2018. “A Gentle Introduction To Linear Algebra”. Machine Learning Mastery. https://machinelearningmastery.com/gentle-introduction-linear-algebra/.
[2] “Best Courses For Linear Algebra”. 2021. helptostudy.com. https://www.helptostudy.com/best-courses-for-linear-algebra/.
[3] Jojoe, Joseph. “You Should Study Linear Algebra.”. 2022. Medium. https://medium.com/@josephjojoe/you-should-study-linear-algebra-eeadd1820946.
[4] “Linear Algebra – Wikipedia”. 2022. en.wikipedia.org. https://en.wikipedia.org/wiki/Linear_algebra.
[5] Kılınç, Çağatay. 2023. “Why Do We Use Matrices For Image Processing?” Medium. Medium. https://medium.com/@cgtyklnc/why-do-we-use-matrices-for-image-processing-3b24a59abe4f.
Matrices are used in image processing for a number of purposes, including storing and manipulating images, applying image filters, and performing image transformations. They are also used in matrix algebra, which is a branch of mathematics that is useful for simplifying complex image processing operations.
In image processing, a matrix is a two-dimensional array of numbers that represents the pixels in an image. Each element of the matrix corresponds to a single pixel in the image, and the value of the element represents the intensity or color of the pixel.
Recommended
“Immersive Math”. 2021. immersivemath.com. http://immersivemath.com/ila/index.html.
An awesome interactive book about Linear Algebra. It’s said to be the world’s first linear algebra book with fully interactive figures. This book includes 10 chapters, explaining all the fundamentals of Linear Algebra (vectors, matrices, linear mappings, eigenvectors and eigenvalues, etc.).
Additional Reading
“1. What Is Linear Algebra?”. 2021. wolfram.com. https://www.wolfram.com/wolfram-u/introduction-to-linear-algebra/what-is-linear-algebra.html.
Ahmed, Mansoor. 2021. “Matrices Role In Linear Algebra”. Technologies In Industry 4.0. https://www.technologiesinindustry4.com/2021/07/matrices-role-in-linear-algebra.html.
Matrices Role In Linear Algebra and play a dominant role in linear algebra. They may be used to efficiently represent systems of linear equations, and then they also represent linear functions or linear mappings. Before we talk over certain of these exciting topics, let us first describe what a matrix is and what kind of operations we can do with matrices.
“An Intuitive Guide To Linear Algebra – BetterExplained”. 2021. betterexplained.com. https://betterexplained.com/articles/linear-algebra-guide/.
Haris, Mohomed Ashkar. “Applications Of Linear Algebra”. 2022. Medium. https://mashkarharis.medium.com/applications-of-linear-algebra-4307a686c194.
The main aim of this papers is to discuss some examples where linear algebra can be used in solving real life problems. This involved modeling real life problem into linear algebraic mathematical structure and solve them using linear algebraic concepts. Within next paragraphs I hope to discuss how linear algebra can be used to build product recommendation algorithm and locate objects in Global Positioning System (GPS). Let us see one by one.
“How I Self-Taught Myself Linear Algebra In A Month (W/O Calculus)”. 2021. Medium. https://medium.com/@abdulhusseinyousifh/how-i-self-taught-myself-linear-algebra-in-a-month-w-o-calculus-8d75c32f1f9e.
I recently complete MIT’s Linear Algebra Course taught by Gilbert Strang (aka 18.06sc). This is my review of its strengths and weaknesses. I will also be giving tips and extra resources you can use.
“Linear Algebra”. 2023. LibreTexts. https://math.libretexts.org/Bookshelves/Linear_Algebra.
Linear algebra is the study of vectors and linear transformations.
“Linear Algebra Part-1”. 2021. Medium. https://medium.com/the-stem/linear-algebra-part-1-84ebe25568a2.
Mathur, Utkarsh. “Matrix & Matrix Operations | Numerical Linear Algebra | Part 1”. 2022. Medium. https://medium.com/@utkarsh.mathur/matrix-matrix-operations-numerical-linear-algebra-part-1-c736a9360e81.
Today I want to talk about Matrix & Matrix Operations.
Mathur, Utkarsh. “Determinant Of A Matrix | Numerical Linear Algebra | Part 2”. 2022. Medium. https://medium.com/@utkarsh.mathur/determinant-of-a-matrix-numerical-linear-algebra-part-2-24c0036bded8.
In the last article I laid some foundations of Matrices and its operations. In this article I’ll discuss the theory of determinants of square matrices.
PolyMaths. “Vectors are NOT what you think they are”. 2023. Medium. https://medium.com/@amitai.rosenbaum/vectors-are-not-what-you-think-they-are-75c75c878bc0.
Here is a seemingly simple question: what is a vector? Most people with a high-school level of mathematics or physics will answer say ‘it’s a quantity which has a direction and a magnitude’. They will compare them to scalars, which have only a magnitude, but no direction. If you ask someone from computer science to give you a definition of a vector, they might respond that it’s similar to a list — a quantity which has coordinates, similar to an n-tuple. Guess what? Both of these definitions are wrong! The ‘real’ definition of a vector is a lot more abstract, but a lot more liberating. The first time I heard the real definition of a vector, I was super excited by the amount of freedom the definition allows. I want to share some of my excitement with you!
Sterling, Mary Jane. Linear Algebra for Dummies. Hoboken, NJ. Wiley Publishing, Inc. 2009.
Strang, Gilbert. Introduction to Linear Algebra, Fifth Edition. Wellesley, MA. Wellesley-Cambridge Press. 2016.
“Why Study Linear Algebra?”. Mathematics Stack Exchange. https://math.stackexchange.com/questions/256682/why-study-linear-algebra.
Videos
“Essence of linear algebra (Playlist)”. 3Blue1Brown. 2021. youtube.com. https://www.youtube.com/playlist?list=PLZHQObOWTQDPD3MizzM2xVFitgF8hE_ab.
“Matrices | Algebra (All Content) | Math | Khan Academy”. 2022. Khan Academy. https://www.khanacademy.org/math/algebra-home/alg-matrices#alg-intro-to-matrices.
Typo correction: At 6:52, the screen shows
[x1, y1] + [x2, y2] = [x1+y1, x2+y2].
Of course, this should actually be
[x1, y1] + [x2, y2] = [x1+x2, y1+y2].
Why is linear algebra actually useful? There very many applications of linear algebra. In data science, in particular, there are several ones of high importance. Some are easy to grasp, others not just yet. In this lesson, we will explore 3 of them: Vectorized code also known as array programming, Image recognition, Dimensionality reduction
Linear algebra is an important tool for engineers and scientists, but that’s not always clear to students who are sitting in linear algebra classes. Here’s the explanation that I think should be at the beginning of your linear algebra class.
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The featured image on this page is from the Linear Algebra video on the YouTube website.
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