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LINEAR ALGEBRA

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  Linear algebra   is the branch of   mathematics   concerning   linear equations   such as: � 1 � 1 + ⋯ + � � � � = � , linear maps  such as: ( � 1 , … , � � ) ↦ � 1 � 1 + ⋯ + � � � � , and their representations in  vector spaces  and through  matrices . [1] [2] [3] Linear algebra is central to almost all areas of mathematics. For instance, linear algebra is fundamental in modern presentations of  geometry , including for defining basic objects such as  lines ,  planes  and  rotations . Also,  functional analysis , a branch of  mathematical analysis , may be viewed as the application of linear algebra to  spaces of functions . Linear algebra is also used in most sciences and fields of  engineering , because it allows  modeling  many natural phenomena, and computing efficiently with such models. For  nonlinear systems , which cannot be modeled with linear algebra, it is often us...
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MATRICES

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  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...
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