Table of Contents - 11.2.6. Multiplication of a matrix with a "column vector"

The module of -dimensional "column vectors" over a ring is the
-dimensional free module over a ring , which is the product of
-many copies of the ring with componentwise addition and multiplication.
Although a "column vector" could also be defined as n x 1 -matrix (according to
Wikipedia "Row and column vectors", 22-Feb-2019,
https://en.wikipedia.org/wiki/Row_and_column_vectors: "In linear algebra, a
column vector [... ] is an m x 1 matrix, that is, a matrix consisting of a
single column of m elements"), which would allow for using the matrix
multiplication df-mamu for multiplying a matrix with a column vector, it
seems more natural to use the definition of a free (left) module, avoiding to
provide a singleton as -dimensional index set for the column, and to
introduce a new operator df-mvmul for the multiplication of a matrix with a
column vector. In most cases, it is sufficient to regard members of
as "column vectors", because
is the base set of , see
frlmbasmap. See also the statements in [Lang] p. 508.