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Let V = R be the set of all reals with the operations (a) u + v = uv and (b) a • u = au. Determine whether V is a vector space or not? Hint: check to see if at least one of the 2 closures or one of the 8 properties fails, then V is NOT a vector space!

Respuesta :

Answer:

Step-by-step explanation:

Let test the axiom of vector space of V

  • Additive axioms:
  1. x + y = y + x Let u, v in V, u + v = uv and v + u = vu, hence it satisfies.
  2. (x + y) +z = x + (y + z), let u, v, w in V (u + v) + w = uv + w and u + (v +w) = u + vw. We can see that [tex](u+v) + w \ne u +(v + w)[/tex] . Hence it is not a vector space.
  3. Note: only 1 of the axiom doesn't satisfy is enough to make a conclusion. we don't need to prove all. However, if it works, we need to prove the next, next, and so on until all axioms work.

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