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.
