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Vector space

Let [a,b] be an interval in {see attachment}. Recall that the set of functions {see attachment} is a vector space over {see attachment} with addition (f+g)(x):=f(x)+g(x) and scalar multiplication

a) choose [a,b]=[0,1]. Decide for each of the following subsets if it is a subspace. Justify your answer by giving a proof or a counterexample: {see attachment}

b) choose [a,b]=[ ]. Show that f(x) = sinx and g(x)=cosx are linearly independent.
?Let u and v be two vectors in a vector space v over {see attachment} . Denote, as usual, by span{u,v} the set of all linear combinations of these two vectors. Show that {see attachment}

Solution Summary

This shows how to determine if subsets are subspaces.