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This paper extends topics in linear algebra and operator theory for lineartransformations on complex vector spaces to those on bicomplex Hilbert andBanach spaces. For example, Definition 3 for the first time defines a bicomplexvector space, its dimension, and its basis in terms of a correspondingvectorial idempotent representation, and the paper shows how an n by nbicomplex matrix's idempotent representation leads to its bicomplex Jordan formand a description of its bicomplex invariant subspace lattice diagram.Similarly, the paper rigorously defines for the first time "bicomplex Banachand Hilbert" spaces, and then it expands the theory of compact operators oncomplex Banach and Hilbert spaces to those on bicomplex Banach and Hilbertspaces. In these ways, the paper shows that complex linear algebra and operatortheory are not necessarily built upon the broadest and most natural set ofscalars to study. Instead, when using the idempotent representation, suchresults surprisingly generalize in a straightforward way to thehigher-dimensional case of bicomplex values.

Bicomplex Matrices and Operators: Jordan Forms, Invariant Subspace Lattice Diagrams, and Compact Operators