Given a matrix $A \in \mathbb{R}^{n\times m}$. Given another column orthonormal matrix $R \in \mathbb{R}^{n\times \ell}$. Now wo do dimensionality reduction of matrix $A$ using $R$ as follows: $$B = R^T A \in \mathbb{R}^{\ell \times m}$$.
My question is the following: Is the row space of $B$ spanned by the same basis as the row space of $A$? (If so, why is this true?)