# How does linear algebra help with computer science?

I'm a Computer Science student. I've just completed a linear algebra course. I got 75 points out of 100 points on the final exam. I know linear algebra well. As a programmer, I'm having a difficult time understanding how linear algebra helps with computer science?

Can someone please clear me up on this topic?

• Linear transformations have many applications in graphics. Mar 28, 2013 at 16:44
• @user10444 can you please give me some examples? Mar 28, 2013 at 16:58
• A nice way to represent graphs is as an adjacency matrix. Many algorithms use this matrix representation and matrix operation to manipulate the graph. Graphs are absolutely fundamental for computer science. Programming is not computer science. Mar 4, 2016 at 2:38

In the second page you read among others

In this class, you will learn the concepts and methods of linear algebra, and how to use them to think about problems arising in computer science.

I guess you have been giving a standard course in linear algebra, with no reference to applications in your field of interest. Although this is standard practice, I think that an approach in which the theory is mixed with applications is to be preferred. This is surely what I did when I had to teach Mathematics 101 to Economics majors, a few years ago.

• Good-to-know. thank-you. Mar 28, 2013 at 21:03

Linear algebra applies to many areas of machine learning. Here is just a small set of examples.

• Support Vector Machines find a best separating hyperplane between two sets of vectors. The optimization problem minimizes an objective function that is most clearly expressed using linear algebra, the minimization algorithms are often solved in the dual space using linear algebra, and proofs regarding the algorithms involve linear algebra.

• Many semi-supervised label propagation graph algorithms can be expressed as optimization of formulae involving the graph's Laplacian matrix.

• Spectral clustering separates data points into groups of related points by finding the eigenvalues of a graph's Laplacian matrix that have small eigenvectors.

• Neural nets use linear algebra in various ways. For example, densely connected neural net layers perform matrix/tensor multiplication to propagate values between them.

• Convex optimization algorithms, which are used throughout machine learning, use linear algebra. The most common algorithm is Low-Memory BFGS.

• Optimization algorithms used for non-convex problems, such as AdaGrad, are often formulated and implemented using linear algebra.

PageRank (which uses stochastic matrices and eigenvectors at its heart) is arguably one of the most useful applications of computer science https://en.wikipedia.org/wiki/PageRank

• From what I've understood, PageRank is just an iterative fixed-point finder, the main idea being that you start with some ranking and then iteratively improve it by convoluting it with the edge weights. You can view this as finding the $1$-eigenvector of a stochastic matrix $A$ iteratively by applying $A$ repeatedly to an arbitrary vector; but this viewpoint is not needed for the implementation! Linear algebra only becomes useful if you want to rigorously analyze the efficiency of the method. If linear algebra wasn't around, PageRank might have easily been invented without it. Aug 22, 2018 at 14:30
• @darijgrinberg yes this iterative procedure is known as the "power method" introduced by von Mises to find eigenvalues and (crucially for PageRank) eigenvectors. I might be in the minority of mathematicians that think of applied math as being a useful language and I sympathize that problem solving is more important than theory building. But can we separate them? Should we ignore the mindset that we obtain by studying foundational topics such as linear algebra in solving problems in tech. The fine tuning of PageRank is a painful hit-and-miss with no Linear Algebra and Perron-Frobenius. Jan 7, 2019 at 10:45

Algebra is used in computer science in many ways: boolean algebra for evaluating code paths, error correcting codes, processor optimization, relational database design/optimization, and so forth.

Matrix computations are used in computer programming in many ways: graphics, state-space modeling, arithmetic, ad hoc business logic, and so forth.

Linear algebra as a sub-discipline is often taught in one of two ways: from a computational aspect of things, which focuses on matrices, their properties, and operations on matrices; or, algebraically, where linear mappings are treated as algebraic structures, and one studies, for instance, the group theoretic relations that arise.

In either case, you will not need to try too hard to find situations where knowledge of either theoretical linear algebra or matrix mathematics will be necessary.

A computer scientist needs various algebraic theories: semigroups, rings, fields, categories. Linear algebra is a base for most of them. Besides, it is used in all other mathematical sciences (differencial equations, probability etc.)