# Vector space property

Tags:
1. Oct 3, 2016

### Math_QED

1. The problem statement, all variables and given/known data

Prove that in any vector space V, we have:

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

2. Relevant equations

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

3. The attempt at a solution

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 $\alpha = 0$

2. Oct 3, 2016

### Orodruin

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

3. Oct 3, 2016

### Staff: Mentor

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

4. Oct 3, 2016

### Math_QED

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

5. Oct 3, 2016

### Staff: Mentor

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.

6. Oct 3, 2016

### Math_QED

Nice. Thanks a lot.