**1** This is an approach often used in control engineering.

**2** All bases contain *exactly N* base vectors. A (putative) alternative base with *M* (*< N*) vectors would imply that there is no set of more than *M* linearly independent vectors – but the original base is just such a set, giving a contradiction. Equally, *M > N* would imply the existence of a linearly independent set with more than *N* members – contradicting the specification for the original base set. Hence *M* = *N*.

**3** It is a useful exercise in close analysis to deduce properties (1.14) and (1.15), on a justified step-by-step basis, using only those given in (1.12) and (1.13) and the general properties of complex conjugation.

**4** Consider a two-dimensional linear vector space in which a typical ...

Start Free Trial

No credit card required