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Algebra: Vecor Spaces

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Determine whether the given set and operations form a vector space. Give reasons.

V = {(x,y,z) : x,y,z are Real numbers},
(x,y,z) + (x',y',z') = (x + x', y + y' z + z'),
k(x,y,z) = (kx,y,z)

b) the set of all positive real numbers with the operations:

x + y = xy, kx = x^k

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Solution Summary

Verifying if two given sets form vector spaces.

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a) Let v=(x,y,z) , u=(x',y',z') two vectors from V and k, k' two scalars.
Then kv+kv'= k(x,y,z) + k(x',y',z') = (kx,y,z) + (kx',y',z') = (kx+kx', y+y', z+z') = ...

Solution provided by:
  • BSc, University of Bucharest
  • MSc, Ovidius
  • MSc, Stony Brook
  • PhD (IP), Stony Brook
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