Linear algebraic systems range

Matrix and tensor notation is useful when dealing with systems of equations. Matrix theory is a straightforward set of operations for linear algebra and is covered in Section A.I. Tensor notation, treated in Section A.2, is a classification scheme in which the complexity ranges upward from scalars (zero-order tensors) and vectors (first-order tensors) through second-order tensors and beyond. 

In the language of linear algebra, N and b define vector spaces, and the dimension of a vector space corresponds to the number of linearly independent vectors, called basis vectors, that are needed to define the space. Then the multiplication in (7.4.2) can be interpreted as a transformation in which A maps a certain subspace of N into a subspace of b. In other words, only certain sets of mole numbers satisfy the elemental balances (7.4.2), and the possible sets of mole numbers depend on the chemical formulae for the species present in the system. That subspace of b, which is accessible to some N, is called the range of A the dimension of the range equals rank(A). According to (7.4.2), any basis vectors for the range automatically satisfy the elemental balances. For example, if we let N represent one particular basis vector for the range, then... 

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