$a*x=0$ implies $x=0$ for all scalars $a$ and vectors $x$.

Question

Assuming $x$ does not equal $0$, assuming $a$ does not equal $0$

$$ax=0 $$ $$ax+ax=0+ax$$ $$2ax=ax $$

Intuitively this is true only for the zero vector leading to a contradiction. How would I show that if two different scalars multiplied by the same vector are equal and equal zero then the vector is $0$.

Answer 1

Suppose $a \ne 0$. Then - as the scalars are from a field - $a^{-1}$ exists. Multiply $ax = 0$ with $a^{-1}$ and get $$ 0 = a^{-1}0 = a^{-1}(ax) = (a^{-1}a)x = 1x = x $$