Explain to a brainlet why Linear Algebra is important

>vectors
>solving systems of equations
>eigenvalue problems
all of these concepts have fuckloads of practical applications and are rooted in linear algebra.

It's a nifty and versatile tool that can be utilized in many areas. Also it's super simple, intuitive and easy.

Have fun trying to do graduate level statistics without linear algebra.

Thanks anons, I'm only in AP Calc II. So all the concepts of Linear Algebra seem pretty abstract to me

Spatial transformations
Quantum mechanics
Derivatives

>This coming from someone who's probably a natural science fag.

It's pretty cool stuff like in the movie The Matrix

Ya have a bunch of systems that are interacting in weird ways, so a bunch of smart cunts worked out how to do bulk algebra on heaps of equations at once using matricies, but using the matrixes comes with a bunch of laws that ya got to follow for it to be mathematically consistent.

So what ya do, is ya take a bunch of nonlinear functions, and you do a bunch of shit to them to make them slightly less accurate, but so that you can quickly evaluate them in matricies at the same time.

you can also do a bunch of maths to work out if your linear system is stable and for what regions it will no longer represent the original system, for instance, for up to 15(deg) when evaluating a pendulum (especially a damped pendulum) sin(theta) = theta and a computer can evaluate this way fcking faster aye

t.control eng who hated maths but loved the applied control theory

Even though I know several generalizations of pretty much every linear algebra concept (kernal, image, feilds, tensors etc) I still end up using base level linear algebra shit when I program.

It's also used as a stepping stool for a lot of babby quantum which is about to change our fucking world.

So yeah, concretely and speculatively, linear algebra is important.

Neural networks are basically just a pile of basic linear algebra that you adjust until it works

Well, the core point of linear algebra is to make certain connections: How matrix algebra (algebra in the ring of matrices) is connected to systems of linear equations, linear transformations, and quadratic forms.

The core part being that you can reduce a statement about linear equations, linear maps or quadratic forms to statements about matrices, which already have a beautiful algebra.

Starting with systems of linear equations, the need to solve them appears everywhere. Including fundamental topics like differential equations. I hope that the importance of being able to solve linear equations efficiently is self-explanatory.

Next linear maps. If you have a linear map and transform it to a matrix, the properties of the matrix become the properties of the map so you can study the matrix itself.

Finally, quadratic forms appear everywhere from geometry to algebra to inequalities. It is therefore very useful to realize that a quadratic form has an associated matrix and again, properties of the matrix become properties of the quadratic form.

Those are in my opinions the main ones. But there are also other objects like Hermitian forms, bilinear forms, multilinear forms, etc. The point of linear algebra is to introduce you to the qualitative study of these objects that dominate mathematics. For example, if you study differential geometry you will study differential forms which arise from multilinear forms.

How are kernel, image, or fields more than basic definitions in linear algebra? I could see modules/rings as generalization, but vector fields are defined over a field to begin with.

t. undergrad freshman in math

Linear algebra is an integral part of multiplication algorithms involving large numbers, particularly those between 2500 digits and 30,000 digits. Using linear algebra takes the algorithm from O(n^2) to O(n^1.465) complexity, potentially shaving off hundreds of thousands of additional computations, provided the inputs are large enough.

See en.m.wikipedia.org/wiki/Toom–Cook_multiplication

they are if you're an engineer

Because it's easier than nonlinear algebra.

They aren't. Did I say they were?

Vector fields are defined over a vector bundle btw, not just a field, if we're going to dick measure generality.

machine learning and 3D graphics you fucking faggot

i don't give an eigenfuck