The product AB means apply B first and then apply A to the result. Equivalently, each column of AB is obtained by applying A to the corresponding column of B, so matrix multiplication represents composition of linear transformations.
Matrix multiplication can be understood as composition, but the same product can also be read column-wise, entry-wise, row-wise and as a sum of outer products.
The chapter starts from geometric composition, derives the computational rule and then shows how the same product can be reorganized into blocks, diagonal actions and special matrix classes.
Each entry cij of C = AB is the dot product of row i of A with column j of B. This entry-wise rule is not a separate definition: it follows directly from the column-wise meaning of matrix multiplication.
The PDF is a single document. The page links below are best-effort: most browsers support them, but some viewers may ignore the page hint.
It shows that a matrix product can be built by adding simple rank-1 layers. This leads to low-rank approximation and later ideas such as SVD.