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(a) Investigate the given question about a set S of functions on an interval I. Give an example. Prove your ansnwer.
(1) If S contains the zero functions, can S be linearly independent?
(2) If S is linearly independent on a subinterval J of I, is it linearly independent on I?
(3) If S is linearly dependent on a subinterval J of I, is it linearly dependent on I?
(4) If S is linearly independent on I, is it linearly independent on a subinterval J?
(5) If S is linearly dependent on I, is it linearly independent on a subinterval J?
(6) If S is linearly dependent on I, and if T contains S, is T linearly dependent on I?
(b) In what cases can you use the Wronskian for testing linear independence? By what means can you perform such a test?