What does "Reflection along the subspace generated by v" means?

Question

I got a problem which includes "Reflection along the subspace generated by $v$ in $\mathbf{R}^{n+1}$". I need some clarification, what does it mean? Does it mean reflection about the hyperplane $v^{\bot}$?

Answer 1

Thanks to Gerry Myerson who guided me to solve this problem. Here is the solution:

let $\mathbf{r}$ be the reflection of the original vector $\mathbf{x}$, along the subspace generated by $\mathbf{v}$. Then both $\mathbf{r}$ and $\mathbf{x}$ has same projection onto $\mathbf{v}$ as reflection along $\mathbf{v}$ will keep the projection onto $\mathbf{v}$ intact.

So, $\mathbf{x}+\mathbf{r}= 2 \times $ (Projection of $\mathbf{x}$ onto the subspace of scalar multiples of $\mathbf{v})= {2 x \cdot v\over \|v\|^2}v\Rightarrow \mathbf{r}={2 x \cdot v\over \|v\|^2}v- \mathbf{x}$