Let M = ( a i j ) 1 ≤ i ≤ m , 1 ≤ j ≤ n {\displaystyle {}M={\left(a_{ij}\right)}_{1\leq i\leq m,1\leq j\leq n}} denote a matrix over a field K {\displaystyle {}K} . Let c = ( c 1 , … , c m ) {\displaystyle {}c={\left(c_{1},\ldots ,c_{m}\right)}} and d = ( d 1 , … , d m ) {\displaystyle {}d={\left(d_{1},\ldots ,d_{m}\right)}} denote two m {\displaystyle {}m} -tuples, and let y = ( y 1 , … , y n ) ∈ K n {\displaystyle {}y={\left(y_{1},\ldots ,y_{n}\right)}\in K^{n}} be a solution of the linear system
and z = ( z 1 , … , z n ) ∈ K n {\displaystyle {}z={\left(z_{1},\ldots ,z_{n}\right)}\in K^{n}} a solution of the system
y + z = ( y 1 + z 1 , … , y n + z n ) {\displaystyle {}y+z={\left(y_{1}+z_{1},\ldots ,y_{n}+z_{n}\right)}} is a solution of the system
denote a matrix over a field 
. Let
and
denote two 
-tuples, and let
be a solution of the
linear system
a solution of the system
is a solution of the system
