Exterior Algebras: Elementary Tribute to Grassmann's Ideas provides the theoretical basis for exterior computations.
It first addresses the important question of constructing (pseudo)-Euclidian Grassmmann's algebras.
Then, it shows how the latter can be used to treat a few basic, though significant, questions of linear algebra, such as co-linearity, determinant calculus, linear systems analyzing, volumes computations, invariant endomorphism considerations, skew-symmetric operator studies and decompositions, and Hodge conjugation, amongst others.
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