Project orthonormal vectors onto subspace while preserving orthonormality

Question

Suppose I have $m$ orthonormal vectors $u_1, ..., u_m \in \mathbb{R}^n$ where $m < n$. I would like to project each vector onto $\mathbb{R}^m$ using some linear transformation $W: \mathbb{R}^n \rightarrow \mathbb{R}^m$ such that $Wu_1, ..., Wu_m \in \mathbb{R}^m$ are orthonormal vectors.

How would I go about finding such a matrix $W$?

Answer 1

It suffices to take $W$ to be the matrix whose rows are $u_1,\dots,u_m$.