So what am I supposed to learn in linear algebra

[Felix Quintana]

I am self teaching the subject but I am unsure of what is the whole point and picture

 

[BvU]

Hard to motivate yourself then. But whatever I write here, it still is external motivation. Take it from me: it's extremely useful.

 

[chiro]

Hey Felix Quintana.

Linear algebra is about generalizing the understanding of things that behave like arrows (think vectors) in terms of the algebra, analysis, synthesis, geometry, and constraints for making use of them.

This is basically the point - to do this using generalized techniques in much the same way that you can do with differentiation and (some) integrals in the standard calculus classes.

 

[BvU]

Turns out there is an awful lot of things that behave like arrows

 

[PeroK]

Linear Algebra is the mathematical basis for Quantum Mechanics.

 

[micromass]

Whenever you meet something that behaves linearly, linear algebra comes into play. What are some things that behave linearly?
- Lines and planes in 3D space
- Lines, planes, hyperplanes in arbitrary dimensional space
- Linear systems of equations
- Linear systems of differential equations
- Geometric transformations such as projection, reflection, rotation

Now this might sound impressive, but it isn't. This is actually a very minor part of mathematics. There are way more things behaving nonlinearly than things behaving linearly. So it might sound that linear algebra isn't all that useful.

Not true at all! The clue is that many things which behave nonlinearly, will have linear approximations. This is why linear algebra arises everywhere. In calculus/analysis, we approximate a curve by its tangent line/plane giving rise to the derivative. In differential geometry, we approximate a manifold by its tangent planes and we approximate its maps by linear maps. We solve nonlinear systems of equations by approximating them by linear systems of equations Etc.

 

[mathwonk]

well that was really an excellent answer to the sentence in your post, i.e. essentially what is linear algebra good for. your title question however seemed to me more of what in linear algebra should you learn, rather than why. so i was going to say, more narrowly, I suggest you should learn the concept of linearity and its consequences for solving (linear) differential equarions. foremost learn the concept of linear dimension of vector spaces, and how it allows you to classify all vector spaces. Starting on linear maps, learn the key concept of the "kernel" of a map and its use in computing general solutions and rank of a map. Then I recommend studying closely the solutions of linear constant coefficient differential equations, as this gives you the model for classifying all linear mappings in finite dimensions. I.e. over the algebraic closure of your scalar field at least, every linear map in finite dimensions looks in some basis like (a sum of copies of) the derivative map acting on the solution set of a linear ode. This is called "Jordan form". Then for applications to Fourier series and geometry, you should learn something about linear maps in the presence of the notion of dot product angle and length, as well as some more abstract forms of "duality", i.e. study especially scalar valued linear functions acting on your vector space. Finally for computations, learn about characteristic polynomials, determinants, and Gaussian reduction of matrices.

In a nutshell, the concept of dimension is fundamental because two vector spaces are essentially the same, i.e. "isomorphic", if and only if they have the same dimension. Moreover two linear maps between the same two (different) vector spaces are the same up to isomorphisms of source and target if and only if their kernels have the same dimension. And two linear maps S,T of the same vector space to itself are the same up to isomorphism of that space, if and only if they have the same characteristic polynomial, and the dimensions of the kernels of all the powers (S-c)^r, (T-c)^r have kernels of the same dimension, where c is any root of the common characteristic polynomial.

 

[Felix Quintana]

well that was really an excellent answer to the sentence in your post, i.e. essentially what is linear algebra good for. your title question however seemed to me more of what in linear algebra should you learn, rather than why. so i was going to say, more narrowly, I suggest you should learn the concept of linearity and its consequences for solving (linear) differential equarions. foremost learn the concept of linear dimension of vector spaces, and how it allows you to classify all vector spaces. Starting on linear maps, learn the key concept of the "kernel" of a map and its use in computing general solutions and rank of a map. Then I recommend studying closely the solutions of linear constant coefficient differential equations, as this gives you the model for classifying all linear mappings in finite dimensions. I.e. over the algebraic closure of your scalar field at least, every linear map in finite dimensions looks in some basis like (a sum of copies of) the derivative map acting on the solution set of a linear ode. This is called "Jordan form". Then for applications to Fourier series and geometry, you should learn something about linear maps in the presence of the notion of dot product angle and length, as well as some more abstract forms of "duality", i.e. study especially scalar valued linear functions acting on your vector space. Finally for computations, learn about characteristic polynomials, determinants, and Gaussian reduction of matrices.

In a nutshell, the concept of dimension is fundamental because two vector spaces are essentially the same, i.e. "isomorphic", if and only if they have the same dimension. Moreover two linear maps between the same two (different) vector spaces are the same up to isomorphisms of source and target if and only if their kernels have the same dimension. And two linear maps S,T of the same vector space to itself are the same up to isomorphism of that space, if and only if they have the same characteristic polynomial, and the dimensions of the kernels of all the powers (S-c)^r, (T-c)^r have kernels of the same dimension, where c is any root of the common characteristic polynomial.

Hey, I just started going over these concepts!, not kernel yet, but thanks! This was very helpful.

 

[mpresic]

Linear algebra is used in several mathematical, physical, and business related areas. It is hard to list all the areas LA touches. Algebra originally developed in a way to solve more and more elaborate equations. Ancients provided the solution to quadratic equations, that you learned in high school. Cubic and quartic (3rd and 4th degree) equations were solved by 15th-16th century Italian mathematicians. Since about 1800, it has been known that the fifth degree equation has no solutions in the form of combinations of radicals.
Algebra has developed along broader (although related) directions since. (I say this because solutions to these equations are associated with symmetry groups.) Linear algebra is a specialized structure (study) within abstract algebra. In electrical engineering it can be used as a tool to solve multi-loop circuits. In physics, it can be used to solve for vibrations in coupled systems. LA is used throughout quantum mechanics.

 

[mathwonk]

the kernel of a linear map T, sometimes called nullspace, is the subspace of the source consisting of all those vectors v such that Tv = 0.

 

[Angelo Marney]

Linear algebra is absolutely central to modern mathematics and physics. I would rate its importance to be just as high as calculus.

For example, vector spaces are one of the first "spaces" that students are formally exposed to.
Many important engineering problems ultimately come down to solving systems of equations (or differential equations).
Multivariate statistics, machine learning, computer graphics, etc are just applied linear algebra.

 

[Felix Quintana]

Interesting! I don't quite understand what the kernel means unfortunately

 

[mathwonk]

did you read post #10? if so and something is not clear ask a question. i.e. v is in the kernel of T if and only if Tv = 0.

aha, an example! the kernel of the derivative operator is the constant functions since Df = 0 if and only if f is a constant (assuming the domain is connected),