  We present a motivating example for matrix multiplication based on factoring
a data matrix. Traditionally, matrix multiplication is motivated by
applications in physics: composing rigid transformations, scaling, sheering,
etc. We present an engaging modern example which naturally motivates a variety
of matrix manipulations, and a variety of different ways of viewing matrix
multiplication. We exhibit a low-rank non-negative decomposition (NMF) of a
"data matrix" whose entries are word frequencies across a corpus of documents.
We then explore the meaning of the entries in the decomposition, find natural
interpretations of intermediate quantities that arise in several different ways
of writing the matrix product, and show the utility of various matrix
operations. This example gives the students a glimpse of the power of an
advanced linear algebraic technique used in modern data science.
