I wish there was a good engineering textbook that presented this clearly! One way is to apply the complex functions to the data before putting it into linear algebra. This is done when curve-fitting data. Another is to use partial derivatives to do a linear approximation of a complex function near a particular point in space. Useful in physics.
Lastly, if you understand properties the matrices and the complex function, you can sometimes combine them directly. So, Linear Algebra is useful any time you have multiple variables. It's useful when you have many moving parts in an engine, concentrations of multiple chemicals in a test tube, many regions of atmosphere in a climate simulation, prices of stocks in a market, etc..
Even though some of those are non-linear systems, there are techniques that may make Linear Algebra useful. They apply transpose to vectors. They focus on calculation rather than use. They focus on irrelevant topics. When you learn ODE, solve some surface integral, do the circuit analysis or numerical analysis, systems control or so you will find it is useful.
You will only find the true value of it when it's the time you relly need it. If you do not tend to a field links to engineering or math, it may not have value. It may like the ancient style prose, when you come to a situation like the ancient says you will find what they have said is ture and moral-philosophical.
Let me give a concrete example. I use linear algebra every day for my job, which entails using finite element analysis for engineering. Imagine a beam. Just an I-beam, anchored at one end and jutting out into space.
How will it respond if you put a force at the end? What will be the stresses inside the beam, and how far will it deflect from its original shape? But now, what if you don't have a straight, simple I-beam? What if your I-beam juts out from its anchor, curves left, then curves back right and forms an S-shape? How would that respond to a force? Well, we don't have an equation for that. I mean, we could, if some graduate student wanted to spend years analyzing the behavior of S-curved I-beams and condensing that behavior into an equation.
We have something better instead: linear algebra. We have equations for a straight beam, not an S-curved beam. So we slice that one S-curved beam into straight beams strung together end-to-end, finite elements.
Beam 2 hangs between beam 1 and beam 3, beam 3 hangs between beam 2 and beam 4, and so on and so on. Each one of these tiny beams is a straight I-beam, so each can be solved using the simple, easy equations from above. And how do you solve simultaneous equations? Linear algebra, of course! Atomic orbitals? Fluid flow? Antenna radiation patterns?
Face recognition? Sign up to join this community. The best answers are voted up and rise to the top. Stack Overflow for Teams — Collaborate and share knowledge with a private group. Create a free Team What is Teams? Learn more. Why study linear algebra?
Ask Question. Asked 8 years, 11 months ago. Active 2 years, 6 months ago. Viewed k times. Aaron Aaron 1, 2 2 gold badges 9 9 silver badges 7 7 bronze badges. Because they like it. Learning how to use linear algebra is to Engineering what ditching your stone-age spear and buying an AK is to hunting!
It is so incredibly useful to a mathematician, and to many applied sciences, there is just no way around it. The answers below are more on topic though. Show 4 more comments. Active Oldest Votes. In other words, if you wanna start thinking, learn how to think straight linear first.
I was trying to figure out how to correct colors from a camera. Turns out that color spaces can be mapped to vectorial space, then solved using linear algebra. Talk about artsy stuff totally unrelated to math. Not just what you described, but for instance when you take a very high-quality file and you want to compress it say in a. When you hear "Fourier transform", think "linear algebra in Hilbert spaces". It's closely related. Signal processing too uses Fourier transforms, and there are a whole load of things in real life that require signal processing.
Systems are described by so called state which is vector and change of state is described by matrices. Than questions about stability becomes questions about eigenvalues. Add a comment. Fit an arbitrary polynomial to some data. I am not even surprised to see so many applications.
If you are impressed with the list you have, just know that for mathematicians, your list is a sub-point in one of my lists. I am not interested at all in applications. Mathematics have the power to make me think of very abstract things which allows me to solve so many problems. At some point, you lose interest in explicitly solving problems i.
Many great mathematicians have used instances from physical sciences as motivation for math, for the sake of understanding. Philosophy concerns the real world, even when it is abstract. Certainly, such a list belongs in the answer of "why study ? Meaning of "Linear" and why it is "Easy" Since you are asking the question, perhaps you would benefit from a discussion of what "linear" means, and why it is "easy", as mentioned in some answers above.
For the mathematician Linear algebra is of course a rich field in its own right but I wanted to write a motivating explanation to aspiring students who do not yet know what it is and how it is relevant to their lives. Ziezi 1 1 gold badge 7 7 silver badges 27 27 bronze badges.
Andreas Andreas 3 3 silver badges 5 5 bronze badges. One application of Linear Algebra is in the use of eigenvalues. Valtteri Valtteri 2, 14 14 silver badges 32 32 bronze badges. Does linear optimizaion still count as Linear Algebra? Thomas Thomas 4, 2 2 gold badges 19 19 silver badges 34 34 bronze badges. Rick Decker Rick Decker 8, 3 3 gold badges 26 26 silver badges 42 42 bronze badges.
I didn't understand Linear Algebra for a while. Your exposition now wants me to take up Linear Algebra. Jebei Jebei 3 3 silver badges 15 15 bronze badges.
I'll replicate simply the top-voted post and its top-voted follow-up comment. AirborneRodent Let me give a concrete example. We have equations for that. A straight, simple I-beam is trivial to compute.
MiffedMouse And to be clear, this kind of situation shows up everywhere. Check Fluid flow? Check Antenna radiation patterns? Check Face recognition? Check Honestly, anything that involves more than one simple element probably uses linear algebra. Therefore, an image is essentially a matrix whose elements are the intensity values of each individual pixel.
To expand, compress, crop or perform any operation on these images, linear algebra is most likely involved. So what is linear algebra? Linear algebra is a branch of mathematics that deals with linear equations and linear functions which are represented through matrices and vectors.
In simpler words, linear algebra helps you understand geometric concepts such as planes, in higher dimensions, and perform mathematical operations on them. It can be thought of as an extension of algebra into an arbitrary number of dimensions. Rather than working with scalars, it works with matrices and vectors. Linear algebra is the building block of machine learning and deep learning.
Understanding these concepts at the vector and matrix level deepens your understanding and widens your perspective of a particular ML problem. For those of you that want a deeper understanding of the inner workings of machine learning and deep learning algorithms, linear algebra is essential.
Khan Academy offers practice exercises, instructional videos, and a personalized learning dashboard that empower learners to study at their own pace in and outside of the classroom. Khan Academy's linear algebra lecture series provides a thorough introduction to linear algebra from the basics of vectors and matrices to projections into lower-dimensional spaces, eigenvectors, and eigenvalues. If you are new to all of these concepts, this is a good place to start! Essence of linear algebra - 3Blue1Brown.
The Essence of linear algebra playlist contains 14 video lectures by Grant Sanderson. The lectures give a geometric understanding of the subject with good visualizations. The lectures cover vectors, linear combinations, matrices, determinants, inverse matrices, systems of linear equations, dot products, cross products, transformations, eigenvalues, and eigenvectors.
If you are a visual learner, these videos are for you! If you are already familiar with the fundamentals of linear algebra, these videos can help you brush up on your basics. Introduction to Linear Algebra - Gilbert Strang. This leading textbook Introduction to Linear Algebra gives a clear introduction to the subject of linear algebra. Unlike most other linear algebra textbooks, the approach is not a repetitive drill and shows the beauty and variety of the subject.
The self-teaching book is loaded with examples and graphics and provides a wide array of probing problems, accompanying solutions.
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