Six-element vector space

Task number: 2508

Is it possible for the structure \((\{0{,}1,2{,}3,4{,}5\},\oplus,\odot)\) to form a vector space over the field \(\mathbb Z_3\), where \(\mathbf u \oplus \mathbf v = \mathbf u+\mathbf v \mod 6\) and \(a\odot \mathbf u= a\cdot \mathbf u \mod 6\)?

  • Resolution

    It is impossible, because for example \(2\odot 3=0\) which is in conflict with axioms of a vector space.

Difficulty level: Easy task (using definitions and simple reasoning)
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