Suppose that $S=\set{\vectorlist{v}{p}}$ is a linearly independent set of vectors in $\complex{m}$. Define the vectors $\vect{u}_i$, $1\leq i\leq p$ by \begin{equation*} \vect{u}_i=\vect{v}_i -\frac{\innerproduct{\vect{u}_1}{\vect{v}_i}}{\innerproduct{\vect{u}_1}{\vect{u}_1}}\vect{u}_1 -\frac{\innerproduct{\vect{u}_2}{\vect{v}_i}}{\innerproduct{\vect{u}_2}{\vect{u}_2}}\vect{u}_2 -\frac{\innerproduct{\vect{u}_3}{\vect{v}_i}}{\innerproduct{\vect{u}_3}{\vect{u}_3}}\vect{u}_3 -\cdots -\frac{\innerproduct{\vect{u}_{i-1}}{\vect{v}_i}}{\innerproduct{\vect{u}_{i-1}}{\vect{u}_{i-1}}}\vect{u}_{i-1} \end{equation*}

Let $T=\set{\vectorlist{u}{p}}$. Then $T$ is an orthogonal set of nonzero vectors, and $\spn{T}=\spn{S}$.