TITLE: Applications of basic linear algebra concepts to computer science?
QUESTION [5 upvotes]: I'm trying to explain linear algebra to some programmers with computer science backgrounds. They took a course on it long ago, but don't seem to remember much. They can follow basic formalism, but none of it seems to stick when they don't see anything in "real life" to relate the concepts to. Do people know any good examples of applications in computer science using basic linear algebra concepts? Some things I was thinking of include:

Eigenvalues and eigenspaces
Image and kernel 
Determinants
Dual space
Transpose
Inner products
Singular value decomposition

REPLY [3 votes]: Spectral graph theory is a big part of theoretical computer science. Some concrete examples are:
1) A big trend in the past decade in TCS has been to utilize the properties of the Laplacian matrix to make classical graph algorithms faster. Usually, the arguments boil down to showing that the problem reduces to solving the linear system $Lx = y$ where $L$ is the Laplacian. Since solving these systems is fast, the resulting algorithms are also fast. Some concrete examples are max flow and graph sparsification.
2) Expander graphs: These are graphs that 'expand' in some way. For example, the number of neighbors of any subset $S$ of vertices is of size at least $c|S|$ for some fixed constant $c$. They 'look like' the complete graph but are much sparser. They have nice spectral properties which means random walks converge fast on them. They are also connected to other branches of TCS through error correcting codes, random graphs, etc.
3) Drawing graphs! It turns out if you have a complicated graph and want to visualize it, a good way to do so is to take the second and third eigenvectors of the Laplacian matrix and use them as $(x,y)$ coordinates for graph vertices. There are some cool examples here. The math behind this is essentially that the eigenvectors are connected to minimizing some form of 'energy' through the Rayleigh quotient.