Comment due Sunday, February 18
You may have heard that matrices are extremely useful tools in several different fields. Our class does not focus on real-world applications of linear algebra, but it might help you slog through some of these long calculations if you have an idea of their potential uses.
Instructions
For this assignment, you will learn a little bit about an application of matrices/linear algebra to a discipline or problem that is interesting to you and/or your major. You don’t have to become an expert (do not worry about being 100% correct here, this is very informal), but you should do your best to answer these questions:
- What is a description of the real-world problem that is being solved using matrices? Can you describe simply it to a non-expert?
- Are there variables? What are the variables called and what real-world quantities do they represent?
- Are there equations relating the variables? Where do the equations come from?
- Are the entries in the matrices coefficients of some variables or do they represent something else?
- Why was this particular application interesting to you?
Summarize your findings as a comment on this post. Your comment should be about 8 to 10 sentences long. Cite your sources and include links to the resources you used. As always, your writing should be your own.
Your comment should be written in a way so that if your classmate is also interested in the same topic, they’ll come away with a bit more of an understanding of how linear algebra is used, together with a link to resources they can check out if they want more information.
Resources
The internet has lots of resources about the application of matrices/linear algebra to different problems/disciplines; some are good and some are not so good, so you will have to use your own judgment. You can always try searching for “applications of linear algebra to [your topic of choice].” Here are a few other suggestions:
- Your textbook describes some applications at the end of section 1.2:
- Balancing Chemical Reactions (chemistry),
- Dimensionless Variables (physics/mechanical engineering),
- Resistor Network (electrical engineering).
- This medium post introduces a few applications of linear algebra to computer science/computer programming, as well as links to resources with more details.
- Linear algebra is an important tool for data science; this article outlines a few uses. Data science and analytics have many applications to problems or disciplines you may be interested in.
Your comment will earn you one participation point.
In the real world, many problems are being solved using matrices. Matrices are a concept where numbers and variables are arranged in a certain way where they form many rows and columns in the shape of a rectangular array (Britannica). An example of a relevant real-world problem that is used today is in economics. People use matrices to study economic status such as trends of shares, businesses, etc. and they use them to create new business models (Embibe). Matrices not only have numbers, but they also have variables such as multivariable calculus functions or derivative operators. These numbers/variables are called entries (Khan Acadamy). When it comes to economics, one example found was this: a one-market equilibrium model where they compare demand and supply (two variables) and two equations (Qs and Qd) with a graph related to them (public.asu.edu), although looking at two different things, both equations show the equilibrium between the two entries and how a one market equilibrium model would look like under certain conditions. This particular application was interesting because the economy is important in today’s society and how certain trends are formed we can learn and understand them better.
SOURCES
Matrix | Definition, Types, & Facts | Britannica
Uses of Matrices: Learn About Application of Matrices in Life (embibe.com)
Matrices, intro (article) | Khan Academy
1 (asu.edu)
The real world problem solved when dealing with matrices is figuring out how much of each variable is in the total. These “variables” represent different parts of a whole. Equations are used to show how much the variables add up to. For example, an equation could be someone having 15 fruits, consisting of apples, bananas, and oranges. If they know they have 5 apples and 2 oranges, how many bananas do they have. When setting up a matrix, the entries would represent the variables. So if a matrix has a row with a 1 2 3, this would mean that you have 1 of the first variable, 2 of the second, and three of the third.While it seems like simple math when put this way, this concept can be applied to many different and more complex real-world problems. An example of this would be chemistry. In chemistry, when a chemical reaction happens, elements are not created or deleted. So when different compounds or elements mix together and form a result, setting up matrices would allow you to accurately calculate how much you originally started with, and how they mixed to form the result. This would show how the elements interact together in the result. The overall topic is interesting in the way that it can be applied to many different fields. It ties these fields together.
The source I used for this was the the Kuttler Textbook
Matrices are used much more in daily life than people would have thought. For example In physics related applications, matrices are used in the study of electrical circuits, quantum mechanics and optics. Engineers use matrices to model physical systems and perform accurate calculations needed for complex mechanics to work.Electronics networks, airplane and spacecraft, and in chemical engineering all require perfectly calibrated computations which are obtained from matrix transformations. A matrix is a rectangular array of numbers and variables for which operations such as addition and multiplication are defined. The entries in the matrices coefiicients generally numbers. The numbers used in matrix may be integer, rational, real, or complex values.This partiular topic was very interesting , the use of matrices in real life, and how it can be applied in all those different fields.
sources: https://medium.com/@singh.saurabh368/application-of-matrices-and-how-it-solves-the-real-life-problems-ad3a3b85c1f7
https://www.embibe.com/exams/where-are-matrices-used-in-daily-life/
https://www.cs.rochester.edu/u/brown/160_10_27_11/lectures/linear_systems/essay_matrices.html
Matrix is a good tool to balance a chemical equation. Dario Sanchez shows an example in the article “Applications of Linear Algebra in Chemistry.”. To balance the equation CH4 + O2 → CO2 + H2O, first give each coefficient a letter value of a, b, c, d: a*CH4 + b*O2 → c*CO2 + d*H2O. After that balance the elements for both sides: carbon: a = c; hydrogen: 4a = 2d; oxygen: 2b = 2c + d. Next, rewrite the three equations in standard form: a – c = 0; 4a – 2d = 0; 2b – 3c – d = 0. These three equations can be written as a 3*5 matrix, which R1 is 1 0 -1 0 0, R2 is 4 0 0 -2 0, and R3 is 0 2 -2 -1 0. We can rewrite the matrix as row reduced echelon form, which R1 is 1 0 0 -1/2 0, R2 is 0 1 0 -1 0, and R3 is 0 0 1 -1/2 0. Then we can get: a = 1/2d; b = d; c = 1/2d. After that, we will let the smallest coefficient equal to 1, then we can get: a = 1; b = 2; c = 1; d = 2. Finally, we can balance the chemical equation as: CH4 + 2*O2 → CO2 + 2*H2O.
Resource: Applications of linear algebra final revise.pdf (utah.edu)
In the field of epidemiology, matrices are used to model the spread of diseases within populations, a pertinent application given recent global health challenges. Variables in these models typically include the number of susceptible, infected, and recovered individuals, represented by S, I, and R respectively. The equations governing these variables stem from differential equations that describe how disease spreads and how individuals move between categories over time. Matrix entries often represent transition rates between states (such as from susceptible to infected), capturing the dynamics of an outbreak. What makes this application fascinating is its ability to combine mathematical precision with public health to inform strategies for disease control and prevention. Understanding these models can provide insights into the effectiveness of interventions like vaccination or quarantine measures. A foundational resource for delving deeper into this topic is the book “Mathematical Epidemiology” by Fred Brauer, which provides a thorough introduction to the mathematical models used in studying infectious diseases.
There are many real-time applications for matrices, and one fascinating application of matrices is in computer graphics, particularly in the field of 3D transformation. We view a 3-D object on a 2D screen, such as in video games or animations, where we’re essentially performing various transformations like rotation, translation, and scaling using Cartesian axes (cartesian refers to describing points in space using coordinates. (Ex: each pixel can be located using Cartesian coordination.) Matrixes are used to manipulate these variables to achieve the desired transformations. For example, to rotate an object. Translation involves adding and subtracting values to coordinate positions. Matrices were originally utilized to solve groups of simultaneous linear equations but were quickly adopted by the computer graphics community to represent the geometric operations for manipulating two- and three-dimensional objects.
Sources:
https://link-springer-com.citytech.ezproxy.cuny.edu/chapter/10.1007/978-1-4471-4321-5_5https://link.springer.com/chapter/10.1007/978-1-4471-0655-5_5
Matrix solutions are widely used in data science to address practical issues. Consider the example of facial recognition. In this case, the variables are the intensities of the pixels that make up an image. The intensity of each pixel is a numerical number that adds together to form a matrix. The correlations between these pixel intensities are modeled by equations, which let the algorithm identify different facial features. The values in matrices are coefficients linked to particular characteristics, such as the contrast between a face’s various portions. As a language model, I find the use of matrices in face recognition fascinating since it bridges the gap between mathematics and practical problem-solving. Matrix applications in data science are consistent with the flexibility and relevance of mathematical ideas in resolving complex issues in various fields.
Reference: https://builtin.com/data-science/linear-algebra-data-science
Topic: Image Processing using Matrices in Computer Vision
Matrices and linear algebra are pivotal in image processing, a vital aspect of computer vision with applications in medical imaging, facial recognition, and satellite imagery. Pixels, representing light intensity in digital images, form the fundamental variables. Matrices represent images, where each element corresponds to a pixel’s intensity. Image convolution, a common operation, involves applying a convolution matrix (kernel) to achieve effects like blurring or sharpening. This mathematical process multiplies image matrix values by corresponding convolution matrix values and sums them. Matrices, acting as filters, determine weights assigned to neighboring pixels, influencing the overall image transformation. This application showcases how linear algebra transforms raw pixel data, allowing for feature extraction, pattern recognition, and computer interpretation of visual information. The extensive use of matrices in tasks like convolution provides a mathematical foundation for enhancing and analyzing digital images, making linear algebra crucial in the realm of computer vision. This particular application interesting to me because I actually took an image processing class where I used MATLAB and we processed different images and used Matrices.
Sources:
https://www.linkedin.com/pulse/matrix-applications-image-processing-shila-kishore
https://www.linkedin.com/pulse/image-processing-matrices-gowri-sathasivam
http://blog.kleinproject.org/?p=588
Matrices are well used in engineering fields such as electrical circuits, structural analysis and control systems. Using linear algebra helps simplify electrical circuits. One of the examples is Kirchoff’s voltage law, which states that in a closed loop sum of voltages equal zero. Let’s say there is two loop circuit with variables and the loop equations are the following R1*I1+R2*(I1-I2)=V and R3*I2+R4*I2-R2(I1-I2 )=0. Above equations contain so many unknown variables. Using linear algebra we can group them in a matrix in such way that R and I variables are split up, and then we can equate them to voltage vector [V, 0] to solve the unknown variables. There are cases where engineers have to deal with more than two circuits, in such cases matrices play crucial role in solving the circuits efficiently. As a math enthusiast and engineering student applications of linear algebra is where two of my most favorite things come together.
Citations:
https://cecs.wright.edu/~krattan/courses/101/labs/lab4.pdf
https://www.linkedin.com/pulse/application-matrices-real-life-mohan-bellu#:~:text=Computer%20Graphics%3A%20In%20computer%20graphics,2D%20screen%2C%20creating%20realistic%20graphics.
https://en.wikipedia.org/wiki/Kirchhoff%27s_circuit_laws
The use of matrices are used in a variety of fields. This includes data scientists as it can improve skills like Machine Learning, Computer Vision, and Natural Language Processing. Machine learning can apply linear algebra with topics such as Loss Functions. Machine learning algorithms intake data, analyze them, and proceed to evaluate the data by applying a given method to the data in order to predict future data. Loss functions are a method of analyzing the accuracy of prediction models. Depending on the accuracy of a given prediction model, the loss function would either produce a high number or a lower number. The lower the number, the more accurate your model is.
Reference:
https://builtin.com/data-science/linear-algebra-data-science
Resistor matrices are one of the methods that can be used to find unknown values is circuits.
Resistor Network Modeling using Matrices are many times necessary, because while simple math can be used for series circuits(components connected in a line where the + terminal on one is connected to the – terminal of another, like older Christmas lights), and parallel circuits(components placed on time of each other, on the same voltage source, like a power strip), it can get more complicated when you have many loops using the combination of the above, while it might be possible to use normal ohms law(V=IR) for many of these complex circuits. It will take too long, so instead you can write down the value of each component, and use a matrix instead.
The variables are usually a Voltage source with many resistors with varying resistance. Capacitors have to be included in AC circuits, as they become resistors at certain frequency’s, so matrix’s representing AC circuits can include formulas that have frequency and farads, that have to be converted to resistance first, in engineering its common to write the formulas in the matrix during setup since its expected to have many changes.
After setting up the matrix as above, and resolving the formulas to just Voltage and Resistance, You use Kirchhoff’s Current Law to solve, all electrical networks have to reach equilibrium, current travels to one place to another to equalize on both sides, since current is only lost though the resistors, the sum of the whole system is 0, so you get the unknowns your looking for, by default its voltage, but you can use ohms law to resolve another type of value like current or resistance.
As a computer engineer I must understand networks of resistors, along with other voltage sources, capacitors and other components to be effective at making subsystems for computers.
While its mainly a component of electrical engineering my major requires it because its multidisciplinary.
Reference:
https://personal.math.vt.edu/embree/cmda3606/chapter2.pdf
My Experience in CET3625(Applied Analysis), and CET3525(Electrical Networks).
Linear algebra plays many important roles in many applications. One application is error-correcting codes, particularly in data transmission. Linear algebra is employed to encode data in a way that allows recovery even if the encoded data is tampered with. One way to do this is to use a generator matrix G to encode each k-bit block of the message into an n-bit block of the code. While I have absolutely no idea what any of these means, they sound cool. This is widely used in DVDs to prevent scratches from affecting movie playback and in deep space probes for reliable data transmission.
Reference: https://www.math.toronto.edu/afenyes/writing/error-correction%20%28october%202015%29.pdf
In my major, we have to analyze circuitry to identify values produced by the circuitry. There are specific circuitry, in this case, mesh circuits where regular analysis is unable to be used. In the case of a mesh circuit, you have 2 options. For the first method, you use differential equations to solve for the unknown values; most of the time the current is your unknown variable. For the second method, you establish matrices based on the system of equations produced by the analysis.
In the case of this circuit, we split it into 2 meshes, which will produce 2 equations based on Kirchhoff’s Voltage Law. We then use the equation in a system of equation which will then put into a matrix. We can perform different linear algebra operations to reach our solutions. You can use the equation A^-1 * B = x, Cramer’s rule, or set up an augmented matrix to find your currents.
Linear Algebra is used in Computer Programming is in Error-Correcting Coding. Put simply, and error-correcting code can be used to retrieve data even if it is has been tampered with. Examples of this application is seen with DVD to prevent scratches on the physical DVD to affect the playback of the movie you want to see. On a bigger scale, it can be seen being used with deep space probes. With the application of error-correcting it is possible for deep space probes to send data back to Earth and keep the sent data intact. As a result, humans have received their first close up pictures of Saturn and Jupiter. Error-Correcting codes seems really interesting to me because it gives new perspective to how we transfer and store data on a day to day basis.
Sources:
https://medium.com/@jeremyjkun/here-s-just-a-fraction-of-what-you-can-do-with-linear-algebra-633383d4153f
In the field of animation and vector design, linear algebra is essential. Linear algebra is responsible for all movement, lighting, and scaling of both 2d and 3d models. For example, let’s say you want to animate a planet going around a star. While you could animate this frame by frame by hand, you can apply a linear transformation, in this case a rotation, to vastly speed this process up. In the world of computer generated lighting, vectors are required for ray tracing. In order to simulate reflections and refractions, one needs to understand how planes intersect and how their normal vectors are positioned. Light will reflect off a plane in the opposite direction of how it enters, which in the case of direct light, is in the direction of the plane’s normal vector. While rendering animation falls more under the umbrella of multivariable calculus, animation would be a tedious slog without the help of linear algebra. As an animator, I definitely use linear algebra on a day to day basis.
Source: https://dl.acm.org/doi/10.1145/1198555.1198795
Matrices from linear algebra have many uses outside of just the classroom. It can also be used for quantum physics concepts, data, computer graphics, animation, and much much more. I enjoy computers since my major is computer science and I find it fascinating how matrices can also be used for computer graphics and animation. It can be used to translate or move objects, rotation, and as well as even scaling. This is interesting to me because it allows us to better handle things more effectively than we would otherwise. They can also be used for texture mapping which makes matrices map 2D texture coordinates to 3D model surfaces accurately. I want to dive more into this sort of field which is why I’m glad matrices from linear algebra are so useful. If you were to make a game, matrices can also handle collision detection and response. So matrices are quite important when it comes to physics specifically.