Deals with the most basic notion of linear algebra, to bring  emphasis on approaches to the topic serving at the elementary level  and more broadly.   A typical feature is where computational algorithms and theoretical  proofs are brought together. Another is respect for symmetry, so that  when this has some part in the form of a matter it should also be  reflected in the treatment. Issues relating to computational method  are covered. These interests may have suggested a limited account, to  be rounded-out suitably. However this limitation where basic material  is separated from further reaches of the subject has an appeal of its  own.   To the `elementary operations' method of the textbooks for doing  linear algebra, Albert Tucker added a method with his `pivot  operation'. Here there is a more primitive method based on the  `linear dependence table', and yet another based on `rank reduction'.  The determinant is introduced in a completely unusual upside-down  fashion where Cramer's rule comes first. Also dealt with is what is  believed to be a completely new idea, of the `alternant', a function  associated with the affine space the way the determinant is with the  linear space, with n+1 vector arguments, as the determinant has  n. Then for affine (or barycentric) coordinates we find a rule  which is an unprecedented exact counterpart of Cramer's rule for  linear coordinates, where the alternant takes on the role of the  determinant. These are among the more distinct or spectacular items  for possible novelty, or unfamiliarity. Others, with or without some  remark, may be found scattered in different places.

Linear Dependence: Theory and Computation PDF

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