Regarding linear regression, in the (informal) solution manual of T. Hastie's "The Elements of Statistical Learning" book, there is a paragraph (p. 16 in the manual) that goes:
"... Recall that we assume that each coordinate of the vector y is independent, and distributed as a normal random variable, centered at $0$, with variance $\sigma^2$. Since these $N$ samples are independent in the sense of probability, the probability measure of the vector $y \in \mathbb{R}^N$ is the product of these, denoted by $N(0, \sigma^2I_N)$. This density has the following form:
$p(\textbf{u}) = \frac{1}{(\sqrt{2\pi}\sigma)^N}exp(-\frac{u_1^2+...+u_N^2}{2\sigma^2})$
where the $u_i$ are the coordinates of $\textbf{u}$ and $\textbf{u}\in\mathbb{R}^N $. Now any orthonormal transformation of $\mathbb{R}^N$ preserves $\sum_{i=1}^{N}u_i^2$. This means that the pdf is also invariant under any orthogonal transformation keeping $0$ fixed. Note that any function of $u_i,...,u_k$ is probability independent of any function of $u_{k+1},...,u_{N}$. Suppose that $\mathbb{R}^N=V\oplus W$ is an orthogonal decomposition, where $V$ has dimension $k$ and $W$ has dimension $N−k$. Let $v_{1},...,v_{k}$ be coordinate functions associated to an orthonormal basis of $V$ and let $w_{k+1},...,w_{N}$ be coordinate functions associated to an orthonormal basis of $W$. The invariance under orthogonal transformations now shows that the induced probability measure on $V$ is $N(0, \sigma^2 I_{k})$..."
How's the assertion above with the bold font that pdf in $\mathbb{R}^N$ induces pdf in $V$ with the same form valid just because $V$ is a subset (with orthogonal basis) of $\mathbb{R}^N$? In other words does this paragraph imply $T: \mathbb{R}^N \rightarrow V$ (assuming some components of $\mathbb{R}^N$ is zero upon transformation, which yields $V$) is an orthogonal transformation just because $\mathbb{R}^N=V\oplus W$? Orthogonal projection transformations can also be done onto an orthonormal base like that of $V$'s and it is known that these projections are not orthogonal transformations in general.
Additionally, what's the deal with coordinate functions? Is this manifold theory?