If $\alpha=0, \beta\neq 0$ and linear combination of $a,b$ is zero, then $b$ is a zero vector?

Question

I'm not sure I understand the following from my textbook:

Vectors $a, b \in V^{2}(0)$ are non-collinear iff $$\alpha a+\beta b=0 \Rightarrow \alpha=\beta=0$$ If there are scalars $\alpha, \beta \in\Bbb R$ at least one of which is not $0$ and that satisfy the equation above, then vectors $a$ and $b$ are collinear.

If $\alpha=0$ and $\beta\neq 0$, then does that mean that $b$ is necessarily a zero vector?

Answer 1

If $\alpha = 0$ and $\beta \neq 0$ and $\alpha a + \beta b = 0$, then yes, $b$ must be zero, simply because

$$\alpha a + \beta b = 0\cdot a + \beta b = 0 + \beta b = \beta b$$

which means that $$\beta b = 0$$

and since $\beta \neq 0$, you can multiply the equation by $\frac 1\beta$ to get

$$\begin{align}\frac 1\beta (\beta b) &= \beta \cdot 0\\ \left(\frac 1\beta \cdot \beta\right) b& = 0\\ 1\cdot b &= 0\\ b&=0.\end{align}$$

If any of the steps above is unclear, please tell so in the comments.