Definition+&+Description+-+Version+2

​ ** Great Job Bertha ans Eduardo... ** An m by n matrix A is a function whose domain is the set of pairs (i,j) were 1<= i <= m and 1 <=j <= n and his ** its ** range is a scalar field commonly denoted by the letter F ( usually the real numbers or the complex nu​mbers ). Seen a matrix as a function may be hard to imagine thats ** that is (do not use contractions in formal papers) ** why we write them as a rectangle with m rows and n columns, inside of w ** h ** ich the element __ubicated__  ** located ** in the i row and the j column is the value of the function A(i,j).

If we have a matrix A whose size is m by n and we have an scalar k that belongs to the scalar field F, then we define de ** extra ** the product between A and k by (kA)(i,j) = k(A(i,j), that is the element (i,j) from the matrix (kA) is the element (i,j) of the matrix A multiplied by the scalar K

If we have two matrices A,B, both of size m by n, then we define the addition between A and B by the matrix (A+B)(i,j)= (A(i,j)+B(i,j)), that is the element (i,j) from the matrix (A+B) is the sum of the elements (i,j) of the matrix A and B.

Matrices have many properties, in order to name them we supose we have 3 matrices A,B,C with m rows and n columns, and we have k,r,s scalars that belongs to the scalar field F we mention before. Some of those properties are the following: 1) A + B = B + A 2) ( (A + B) + C ) = ( A + ( B + C) ) 3) k ( A + B ) = k A + k B 4) (r + s) A = r A + s A