Have been talking up learning linear algebra to some of the programmers at work (in particular, using Gil Strang’s fantastic lectures), and am now looking for projects to work with them at various levels. In particular, any interesting programs/scripts/projects that might only utilize matrix subspaces.

Excited to move on to perhaps interactive curve fitting once they cover least squares, PageRank when they get to eigenvalues, and maybe image compression using singular values. Just having a hard time coming up with a good project that *actually* starts from matrix subspaces (i.e., not casing an eigenvalue as a mysteriously useful null space value).

A great end result of all this might be a project investigating large networks and locating clusters.

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Linear error-correcting codes, such as Hamming(7,4), don’t use anything about matrices except null / range subspaces (over a two-element field).

This is great! Likely I can suppress everything about this being over a finite field as programmers are already pretty sharp with addition mod 2… (at the very least, through the XOR operator)

The (i,j)-th entry A^k, the k-th power of the adjacency matrix of a labeled graph, tells you exactly the number of length k walks from node i to node j. That only uses matrix multiplation. Also, the determinant of any cofactor of a graph laplacian tells you the number of spanning trees in the graph. That uses matrix subspaces and being able to compute a determinant. Finally, the number of connected components of a graph (i.e. number of clusters) can be determined as the nullity of the graph laplacian. Rank-nullity theorem!