Two related matters are omitted.
Firstly, the Sylvester normal form, or closely related Spectral, characterizations and proofs of the rank-nullity theorem. These are omitted because our rank-nullity chapter is placed prior to either normal forms or Eigentheory ( = Spectral Theory) making their first major appearance. So, were the above converted into a longer Review, then these simpler aspects would be placed in the text prior to Naming Remark 4 and 5’s discussion.
Secondly, discussion of more extended list forms of fundamental theorem of Linear Algebra maps +. This is omitted since one of the requisite ingredients is the diagonal decomposition (DD), more widely and confusingly known as singular-value decomposition (SVD), of matrices. And this is also in a later chapter. The other ingredients – direct-sum dceompositions and Gram-Schmidt orthonormalization – are already in play. The first is accorded an exercise, while the second is judged to go better in the discussion section after DD. Which centres on the extent to which hitherto proposed FuToLAM+ extend from the reals and the complex numbers to arbitrary fields. While this itself has Functional Analysis and Abstract Algebra extensions, those are advanced enough to only feature as open exercises in Parts X and XI of the Book (in contrast, the current Chapter 17 is the last chapter in Part III).
Concepts of Shape 