Let $A$ be an $r$-rank, $m\times n$ matrix with singular value decomposition $A=UDV^T$. Let $T(A)$ be the matrix linear subspace $$T(A)=\{UY^T+XV^T\mid X\in\mathbb{R}^{m\times r}, Y\in \mathbb{R}^{n\times r}\}$$
Is the orthogonal projection [with respect to the Euclidean inner product $\langle A,B\rangle= tr(A^TB)]$ of $A$ onto $T(A)$ equal to $UV^T$?