Vector space basics: scalar times a nonzero vector = zero implies scalar = zero?

An idea for you on the same varation: suppose the scalar $\;a\in\Bbb F\;$ is not zero. It then has an inverse $\;a^{-1}\;$, and:

$$av=0\implies a^{-1}(av)=a^{-1}0\implies 1\cdot v=0\implies v=0$$

The above assume you already know that scalar times the zero vector equals the zero vector...


Hint: There was that axiom that said that $1v=v$, was there not?

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Linear Algebra

Axioms

Proof Verification

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