The most usual representation of systems on computers and elsewhere is using matrices. Finite element problems, dynamics, fluid mechanics, ..., are almost always matrix problems for the computer.
A matrix A is a table of numbers:
matrix consists of n column vectors (the columns), or
equivalently of m row vectors (the rows).
Conversely, a column vector is equivalent to a matrix with only one column and a row vector is a matrix with only one row.
Square matrices are matrices with the same number of rows as columns.
Index notation:
indicates ``the collection of values'' or ``set of
values''.