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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 Preview

Please see attached file.

1.a)

(1) Let A={f: f(1)=0}
First (f+g)(1)=f(1)+g(1) = 0+0=0, so (f+g) A
Secondly , so A
Therefore A is a subspace.

(2) Let B= {f: f(1)=2}
First (f+g)(1)=f(1)+g(1) = 2+2=4 , so (f+g) B, and therefore B is not a subspace because addition ...

Solution Summary

This shows how to determine if subsets are subspaces. Vector spaces are analyzed. A counter example is provided.

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