 Solution

In the column convention used for QR factorization, an $m\times k$ real matrix has orthonormal columns when $Q^{T}Q=I_k$. This says that each column has unit norm and distinct columns have zero inner product. If $Q$ is square, then $Q^{T}Q=I$ provides a left inverse; a square matrix with a left inverse is invertible, so $Q^{-1}=Q^T$, and also $Q{Q}^T=I$. Such a square matrix is an orthogonal matrix. A rectangular matrix with orthonormal columns generally has no two-sided inverse.