Rodriguez Tucker
10/01/2023 · Primary School
(5) Let \( V \) be the set of all Pairs \( (x, y) \) of real numbers and \( F \) be the field of real numbers Define, \[ (x, y) \oplus\left(x, y_{1}\right)=\left(x+x_{1}, y+y_{1}\right) \] \[ a \odot(x, y)=(a x, y), a \in \mathbb{R} \] show that \( V \), with these operations, is not a Vector space over the field of real numbers.
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The operations defined on \( V \) do not satisfy the distributive property of scalar multiplication over vector addition, so \( V \) is not a vector space over \( \mathbb{R} \).
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