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Each of the following sets spans a subspace of the complex vector space of all functions from C to C. (Here the scalar field is C, not R.) In each case find a basis for the subspace and prove it is a basis; state the dimension.

exp iz; cos z;sin z

exp iz; cosh z;sinh z

i posted a similar question before, but this is in the Complex field and not real. someone please explain how to prove this?

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In each case, your basis will be a subset of the functions you are given. If all of the functions are linearly independent, then your basis is just all of the functions. If the functions are not linearly independent, eliminate functions one by one until you get a linearly independent set. – rayradjr Nov 28 '12 at 23:03

Do you know that

$$\sin z=\frac{1}{2i}\left(e^{iz}-e^{-iz}\right)\;\;,\;\;\cos z=\frac{1}{2}\left(e^{iz}+e^{-iz}\right)$$

The above must make question $1$ almost trivial

For the second question just take into consideration the relations

$$\sinh z=i\sin iz\;\;,\;\;\cosh z=\cos iz$$

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