Explain Linear Algebra to someone who has never taken it before

Linear algebra is the branch of mathematics concerning vector spaces and linear mappings between such spaces. It includes the study of lines, planes, and subspaces, but is also concerned with properties common to all vector spaces.

>>7667204
It's also solutions to systems of equations.

>>7667200
So in regular high school algebra you manipulate one equation at a time. In matrix algebra you manipulate lots of equations all at once. Then there's some theory tacked onto it about applications of these matrix functions to different mathematical concepts like coordinate systems and vectors that it turns out are all actually inter-related, and all governed by applying lots of regular algebra equations all at once.

>>7667200
Linear Algebra = Algebra in higher dimensions

Vector spaces over some field F, also called linear spaces, are basically sets that contain ax+by whenever they contain x and y for any a,b in F, with a few extra restrictions that define how they work algebraically. The x-y plane is a simple example

We call elements x_1, ... x_n and y linearly independent if (a_1)(x_1)+...+(a_n)(x_n)=/=y for a in F. We call elements x_1, ... x_n, ... generating if for every element y, y=(a_1)(x_1)+...+(a_n)(x_n)+... If a system of elements is both generating and linearly independent we call it a basis, and the number of elements in the basis is called the dimension of the space, which may be either finite or infinite. You can choose an infinite number of different bases for the same space, and each will have the same number of elements, so dimension is an intrinsic property of a linear space. Linear algebra cares more about finite dimensional vector spaces.

From there, you look at finite dimensional linear forms, i.e. functions A such that A(ax+by)=aA(x)+bA(y) with x,y being elements of a finite dimensional space and see how they can be represented as matricies for different choices of basis (note aside that the set of all mxn matricies is itself a vector space)

Other elementary topics include change of basis, the jordan canonical form of a linear operator, bilinear and quadratic forms, inner product and unitary spaces, and maybe a formulation of the tensor concept.

Linear Algebra = your ticket into higher dimensional space.

>>7667599
Plus it's really useful for creating 3D games among other things.

Eigenvalues, eigenvalues everywhere.
And
IT AIN'T GONNA INTEGRATE ITSELF

>>7667289
retard spotted

Learn how to gaussian algorithm and you will easily pass

I'm finishing up a semester of linear algebea. What ive learned so far is Gaussian elimation and a linear transformation is injective iff N(T) = {0}

in the vectorial space no one can hear you scream

>>7668902
Eigenvalues and Eigenvectors?
Gram Schmidt Orthogonalization?
Invertible Matrix Theorem?
Hell, even what a Basis is?

File: down-syndrome-work.jpg (28KB, 615x410px) Image search: [Google]

28KB, 615x410px

>>7668919

I was just messing friend :)

>>7667200
linear algebra is so boring. literally the point where I quit STEM.

>>7669094
Linear algebra is just a tool for scientists/mathematicians
Learning how to use it isn't supposed to be that interesting, fag

>>7669094
>>7669106
Pleb taste.

>>7667200
Turns out you can change coordinates and still describe your vectors man. Fun subject, linear algebra.

In short, it is the study of vector spaces, ie. sets of things that you can add and multiply by a real (incidentally, you can do that with vectors in the plane, but also with all sorts of other objects such as functions, polynomials and other things)
It allows a formalization of the notion of independence and dimension (number of independent numbers necessary to describe any element of your set).
You can then define linear maps, ie. maps that respect the vector space operations.
It is useful because it is simple and appears in pretty much all parts of mathematics. It is the perfect example of a good generalization (a notion general enough to appear in a wide number of mathematical subjects but not so general that nothing nontrivial can be said about it)