Orthonormal Sets and Matrices

A set of vectors $\{\mathbf{u}_1, \ldots, \mathbf{u}_n\}$ is orthonormal if every vector is a unit vector and every pair is orthogonal: $\mathbf{u}_i \cdot \mathbf{u}_j = 0$ for $i \neq j$ and $\mathbf{u}_i \cdot \mathbf{u}_i = 1$. The standard basis vectors $\mathbf{e}_1, \ldots, \mathbf{e}_m$ are the simplest example.

If we stack $n$ orthonormal vectors in $\mathbb{R}^m$ as the columns of a matrix $U$, then $U^t U = I_n$. The $ij$-entry of $U^t U$ is $\mathbf{u}_i \cdot \mathbf{u}_j$, which is $1$ when $i = j$ and $0$ otherwise. This means $U^t$ is the left inverse of $U$.

Because $U^t U = I_n$, the columns of $U$ are linearly independent — the transpose gives you coordinates for free, with no solving required. Any subspace has an orthonormal basis, obtainable via the Gram-Schmidt algorithm.