I'm getting confused between 2 variants of definition of induced matrix norm. Given a norm $||\cdot||$ on $\mathbb{R}^n$, the induced matrix norm is defined by $$ \left\lVert A \right\rVert = \max_{\mathbf v\not =0}\frac{\left\lVert A\mathbf v \right\rVert}{\left\lVert \mathbf v \right\rVert} \qquad for \quad A \in \mathbb{R}^{n\times n}.$$
I'm trying to deduce the second variant from this definition i.e. $$\left\lVert A \right\rVert =\max_{\Vert \mathbf w\Vert = 1}\Vert A \mathbf w\Vert.$$
Consider $\mathbf v= \frac{\left\Vert\mathbf v\right\rVert\mathbf v}{\left\Vert\mathbf v\right\rVert}=\mathbf w\left\Vert\mathbf v\right\rVert $ where $\mathbf w= \frac{\mathbf v}{||\mathbf v||}$ and $||\mathbf w||=1$.
Therefore, $$ \left\lVert A \right\rVert = \max_{v\not =0}\frac{\left\lVert A\mathbf v \right\rVert}{\left\lVert \mathbf v \right\rVert}= \max_{v\not =0}\frac{\left\lVert A(\mathbf w\left\Vert\mathbf v\right\rVert )\right\rVert}{\left\lVert \mathbf v \right\rVert}=\max_{\mathbf v\not=0}\frac{||\mathbf v||}{||\mathbf v||}\Vert A \mathbf w\Vert.$$ I don't know how to continue from here on.
Also, I 'm reading textbooks where they say : $$\left\lVert A \right\rVert =\max_{\Vert \mathbf w\Vert = 1}\Vert A \mathbf w\Vert \quad for \quad \mathbf w \in \mathbb{R}^n .$$
My question is shouldn't it be $$\left\lVert A \right\rVert =\max_{\Vert \mathbf w\Vert = 1}\Vert A \mathbf w\Vert \quad for \quad \mathbf w \in \mathbb{R}^n \setminus \{\mathbf 0\} .$$
Because $\mathbf w=\mathbf 0$ means $||\mathbf w||=0$ which is ruled out because $||\mathbf w||=1.$