Proving Vector Space Property: αa = 0 ⟹ α = 0 or a = 0

[member 587159]

Homework Statement

Prove that in any vector space V, we have:

$\alpha \overrightarrow a = \overrightarrow 0 \Rightarrow \alpha = 0 \lor \overrightarrow a = \overrightarrow 0$

Homework Equations

I already proved:

$\alpha \overrightarrow 0 = \overrightarrow 0$
$0 \overrightarrow a = \overrightarrow 0$

The Attempt at a Solution

[/B]
Suppose $\alpha \neq 0$

Then: $\alpha \overrightarrow a = \overrightarrow 0$
$\Rightarrow \alpha^{-1} (\alpha \overrightarrow a) = \alpha^{-1} \overrightarrow 0$
$\Rightarrow (\alpha^{-1} \alpha) \overrightarrow a = \overrightarrow 0$
$\Rightarrow 1 \overrightarrow a = \overrightarrow 0$
$\Rightarrow \overrightarrow a = \overrightarrow 0$

The problem is. I don't know how to show that $\alpha \overrightarrow a = \overrightarrow 0$ can imply $\alpha = 0$ I can't suppose $\alpha = 0$, because I need to prove that?

Maybe something like this?

$\alpha \overrightarrow a = \overrightarrow 0$
But $0 \overrightarrow a = \overrightarrow 0$

Thus: $\alpha \overrightarrow a = 0 \overrightarrow a$

Comparing the two, we obtain $\alpha = 0$

Thanks in advance.

 

[Orodruin]

I suggest you focus on the implications of assuming that $\vec a \neq 0$.

 

[fresh_42]

You can distinguish cases. Either $\alpha = 0$ or $\alpha \neq 0$. One of the two has to happen.

 

[member 587159]

You can distinguish cases. Either $\alpha = 0$ or $\alpha \neq 0$. One of the two has to happen.

But can I assume $\alpha = 0$? Then it follows trivially that $\alpha = 0 \land \alpha \overrightarrow a = \overrightarrow 0 \Rightarrow \alpha = 0$

 

[fresh_42]

But can I assume $\alpha = 0$? Then it follows trivially that $\alpha = 0 \land \alpha \overrightarrow a = \overrightarrow 0 \Rightarrow \alpha = 0$

Why not?
$$A = A \wedge \text{ true } = A \wedge (B \vee \lnot B) = (A \wedge B) \vee (A \wedge \lnot B)$$
and $B=(\alpha = 0)$ does the job.

 

[member 587159]

Why not?
$$A = A \wedge \text{ true } = A \wedge (B \vee \lnot B) = (A \wedge B) \vee (A \wedge \lnot B)$$
and $B=(\alpha = 0)$ does the job.

Nice. Thanks a lot.