How can I prove that the set of all planar vector fields forms a vector space? Thanks for any input!
Welcome to PF;
You need to start with the definition of a planar vector field - how would you tell if a particular field were a member of the set?
Then you need to compare this with the definition of a vector space ... which operations on planar vector fields would correspond to the different vector operations.
Presumably you've already seen how to do this with some examples that don't seem, at first glance, to be vectors ... like polynomials?
if V is a vector space and S is a set, then the set of functions S-->V is a vector space.
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