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I am having trouble understanding what this all means for sets other than [tex]F^n[/tex] I find vector space and subspace to be intuitive concepts when applied to complex number sets (particularly so with respect to real number sets). But I have trouble understanding what vector space and subspace actually means when applied to sets containing non-numeric elements. For example, Axler discusses subspaces in the context of the set P(F), which is the set of all polynomials with coefficients in F, and the function p(x), which is a polynomial function. The example he provides is that the set [tex]\{\rho \in P(F) : \rho(3) = 0\}[/tex] is a subspace of [tex]P(F)[/tex]

This makes a bit of sense to me, but I'm having trouble understanding its significance. I don't get why the concept of subspace is useful for anything other than vector space having ordered n-tuples as elements. I'm having similar difficulty with the concept of direct sum, but I'll save that until after I've cleared up my current confusion.

Thanks.