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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 ...

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