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}} be an m × n {\displaystyle {}m\times n} -matrix over a field K {\displaystyle {}K} . Show that the corresponding mapping
is linear.