What problems can we solve using linear algebra?

Question

I thought that linear algebra is a tool for solving systems of linear equations, but this can be done without most of linear algebra. That is, we just have to know matrix and the gaussian elimination and we don't have to know vector space, linear map, determinant, dimension, etc...

If we learn linear algebra, what problems can we solve other than systems of linear equations?

Answer 1

In my numerical linear algebra class my professor stated that 70% of problems can be summarized as the $Ax=b$ problem and the eigenvalue problem. The rest of the problems are slight variations of these problems.

There are a lot of applications. My advisor worked on stabilizing a method for solving hyperbolic PDEs and later he made them faster using FFTs and the SVD.

If you look at most websites like for instance Facebook, Amazon, Netflix or other places they use recommender systems based on linear algebra. These use the SVD algorithm to find users and items that are near other users or items and recommend them to other people.

Answer 2

Linear Systems, Manifold Systems or systems that are almost linear. Hence, you are speaking about systems in which the configuration space is either a hypersurface i.e $\mathbb{R}^N$ for some $N$ or a manifold $M^N$ where $N$ denotes the dimension. Lastly, by almost linear this means you don't have to be a manifold but the underlying data set relates strongly enough to define tangent planes i.e linear maps.

Answer 3

My favorite application of linear algebra is in computer graphics. Using eigenvalue decomposition allows for quick computation of large powers of matrices.

In computer graphics, things like 3D models and plants are modeled using triangles all stuck together, and then they're smoothed out using a process involving taking large powers of matrices. Linear algebra makes this very quick, which allows us to play 3D games at a reasonable rate!!

For more description, see this video :)

Answer 4

There was a comment years ago to the effect of "Why do we study linear algebra? Because linear algebra is easy and many problems can be reduced to it." Solutions to homogeneous differential equations form a vector space as do solutions to homogeneous recurrence relations. Coupled differential equations become much simpler if you change the basis to decouple them. All of Fourier analysis is essentially a change of basis in the vector space of functions. The fact that $\int (af(x)+bg(x))dx=a\int f(x)dx+b\int g(x)dx$ reflects that the integrable functions form a vector space.