MATRICES
Matrices [ edit ] Main articles: Matrix and Determinant A typical matrix Matrices are a useful notion to encode linear maps. [22] They are written as a rectangular array of scalars as in the image at the right. Any m -by- n matrix � gives rise to a linear map from F n to F m , by the following � = ( � 1 , � 2 , … , � � ) ↦ ( ∑ � = 1 � � 1 � � � , ∑ � = 1 � � 2 � � � , … , ∑ � = 1 � � � � � � ) , where ∑ denotes summation , or, using the matrix multiplication of the matrix � with the coordinate vector � : � ↦ � � . Moreover, after choosing bases of V and W , any linear map f : V → W is uniquely represented by a matrix via this assignment. [23] The volume of this parallelepiped is the absolute value of the determinant of the 3-by-3 matrix formed by the vectors r 1 , r 2 , and...