Matrices are a useful notion to encode linear maps. They are written as a rectangular array of scalars as in the image at the right. Any m -by- n matrix A gives rise to a linear map from F to F, by the following x = ( x 1 , x 2 , … … , x n ) ↦ ↦ ( ∑ ∑ j = 1 n a 1 j x j , ∑ ∑ j = 1 n a 2 j x j , … … , ∑ ∑ j = 1 n a m j x j ) , where ∑ ∑ denotes summation, or by using the matrix multiplication of the matrix A with the coordinate vector x :