Seminorms on E_n

Abstract

In previous chapters, we have regarded the relation:

$$\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{A}\otimes\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{x}=\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{b};\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{A}\varepsilon{{M}_{mn}};\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{x}\varepsilon{{E}_{n}};\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{b}\varepsilon{{E}_{m}}$$

(18-1)

as embodying firstly a set of simultaneous equations and secondly a linear dependence among m-tuples. In the present chapter we shall examine (18–1) in yet another light and regard it as expressing the fact that m-tuple b belong to a certain space generated by the columns of matrix A.