Let $\alpha \in \mathbb{R}^n$, and let $s_{\alpha}: \mathbb{R}^n \rightarrow \mathbb{R}^n$ be the map that reflections each point around the hyperplane that is normal to $\alpha$. We can write out the effect of this map explicitly:
$s_{\alpha}(b)=b - 2 \frac{(b,\alpha)}{(\alpha,\alpha)}\alpha$
I have come a long way in understanding why this formula is what it is thanks to you guys! I am now wondering what would be the geometric significance of this formula if we defined it by:
$s_{\alpha}(b)=b - \frac{(b,\alpha)}{(\alpha,\alpha)}\alpha$
That is, we only subtract one projection of $b$ onto $\alpha$ instead of two. What would this map be describing? What would the image of specific vectors look like? Thanks !!