How do I multiply a 0 vector by the inverse of a scalar?

Question

So I need to prove that: Let be a vector space, let ⃗ ∈ , and let ∈ R. Prove that ⃗ = $\vec0$ implies = 0 or ⃗ = $\vec0$. I am supposed to start off the proof using the information: If ≠ 0, then multiply both sides of the equation ⃗ = $\vec0$ by 1/.

I understand that ⃗ * 1/ results in ⃗, but what does $\vec0$ * 1/k equal? I know a zero vector works as an additive identity but I don't know how it affects the scalar inverse when multiplied. Any advice is appreciated.

Answer 1

Multiplying the zero vector by anything results in the zero vector. You can deduce this from the fact that it is the additive identity.

$\vec{0} + \vec{k} = \vec{k}$ for any vector $\vec{k}$. So $\vec{k} - \vec{k} = \vec{0}$ for any vector $\vec{k}$. Therefore, $a(\vec{k} - \vec{k}) = a\vec{0} \Rightarrow a\vec{k} - a\vec{k} = a\vec{0}$. But $a\vec{k}$ is just another vector, so $a\vec{k} - a\vec{k}$ must also equal $\vec{0}$. Therefore, $a\vec{0} = \vec{0}$.

Not sure how much information you would be allowed to use if you must prove this fact, as I am guessing you are relatively new to linear algebra. Hopefully this helps you understand the idea though :)

Answer 2

The inverse of a (nonzero) scalar is just another scalar; hence

$$\frac 1 k \vec 0 = \vec 0$$

If you're not aware of this fact - that, more generally, $a \vec 0 = \vec 0$ for any scalar $a$ - I suggest proving it.