i know that the set "all real numbers" make up a vector space, but how do you prove that it is so?
I would think that you would have to show that vector addition and scalar multiplication are defined on the set, subject to conditions.
Over what field....
add up two real numbers, get a real number, multiply a real number by a real number get a real number, has a zero vector, therefore it's a vector space, over R, obviously 1-dimensional.
Equally obviously it is therefore a vector space over any subfield of R, not necessarily 1-d.
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