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Now that you are familiar with vector spaces, try determining if other sets of objects meet the requirements (or not). For example, the set of integers, the set of rationals, the set of positive continuous functions on a closed interval. Do these meet the definition of a vector space? If not, why? Can you think of other examples from your past?
Not a stupid question. Are the integers over the integers a vector space?
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on this one, note that you must have a field as the basic algebraic structure. The integers are not a field. Hence the integers over the integers are not a vector space.
What about the rational numbers over the rationals?
lol ya it does
that was a wild guess
yes, the rationals are a field.
y is that so?
Fields is a set together with two operations: addition + and multiplication x, which satisfies just about every axiom of the real numbers. Such as there exists a number 0 such that
0 + x = x for all x.
The rationals behave in the right way.
Anyway, let's stick with the real numbers. The finite polynomials are a vector space over the reals also.
If you go hunting on the internet, you'll find lots and lots of examples.