Gaussian distribution propoprties

Good day,

I've a question regarding Gaussian distribution properties over lattices :

Let $$\mathcal{L}$$ := $$\mathcal{L}(\,b_{1}$$,..., $$b_{m})$$ be a lattice over $$\mathbb{R}^{n}$$, and $$W$$ = span($$b_{1}$$,..,$$b_{m}$$)$$^{\perp}$$. define $$\pi_{W}$$ to be the orthogonal projection onto $$W$$.

If i sample a vector b from a Gaussian distribution of support $$\mathcal{L}$$, standard deviation parameter $$s$$ and center parameter $$c \in$$ span($$\mathcal{L}$$).

Can i pretend that $$\pi_{W}(\,b)\,$$ can be sampled from a Gaussian distribution of support $$\pi_{W}(\,\mathcal{L})\,$$, standard deviation parameter $$s$$ and center parameter $$\pi_{W}(\,c)\,$$?

Thanks

• Am I missing something? Surely $\pi_W(b)$ is the zero vector? Commented May 15, 2023 at 14:55
• Certainly it is. but, what i was trying to say that i didn't expressed very well- and I'm really sorry about it-; is how the Gaussian distribution handle linear transformation such as orthogonal projections. Commented May 16, 2023 at 11:20