Let $W \in \mathbb{R}^{n \times m}$ and $P_W$ be the projection onto a linear subspace $span(W) \subset \mathbb{R}^n$. In this case, is $P_W$ the projection onto the row space or the column space of $W$?
When I see $span(W)$, I interpret it as the span of the vectors in $W$. However, is this referring to the span of the row vectors (row space) or the column vectors (column space)?
Best Answer
Note that the rows of $W$ live in $\mathbb R^m$ and the columns live in $\mathbb R^n$, so $P_W$ must be projecting onto the column space of $W$.
(IMO, it's best to use notation $\text{Span}(S)$ only when $S$ is a set of vectors.)