We turn the Gram-Schmidt process into a new factorization $\mathbf{A}=\mathbf{Q}\mathbf{R}$. We're thinking about the columns of $\mathbf{A}$ being a basis for column space ($r=n$), and we want to produce an orthonormal basis. The matrix $\mathbf{Q}$ is the same shape as $\mathbf{A}$ and $\mathbf{R}$ is square, a combining matrix of sorts.