How does linear algebra help with computer science?

The page Coding The Matrix: Linear Algebra Through Computer Science Applications (see also this page) might be useful here.

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.

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