Prove that if x is a vector and a is a scalar, then the following relation holds ? 1) if ax = 0, then either a = 0 or x = 0 ( or both). This is trivial although i am unsure if my steps are correct.
step 1 Lets pick $a = 0$ and $x = (x_1, x_2,\ldots, x_n)$ where for all $x_1,\ldots, x_n$ are not zero. $ax = ax_1 + ax_2 + \cdots + ax_n$ as multiplication by scalars is distributive. Now can I just state $ax = ax_1 + ax_2 + \cdots + ax_n = 0$ as a is zero every where there doesn't seem to be any axiom of vector space which i could quote as reasoning there or is there ?
step 2 Lets pick $a\neq 0$ and $x = (x_1, x2,\ldots, x_n)$ where for all $x_1,\ldots x_n$ are zero. Again as before $ax = ax_1 + ax_2 + \cdots + ax_n$ as multiplication by scalars is distributive.
I know there is a zero vector in the vector space and according to my assumption $x$ is a zero vector but how do i justify $ax = 0$ ?
Any help or guidance would be highly appreciated.