What is a particular use of Gram-Schmidt orthogonalization? We have a linear space V of $m \times n$ matrices. I know that we can use Gram-Schmidt to construct an orthonormal basis, but the natural basis for this space (where every ij-th element is $1$ and the rest $0$) is just that - every matrix there is orthogonal to the rest, and each norm equals $1$.
Where does the algorithm come into use? Why would somebody go through the trouble of constructing a new basis when the natural one fits the bill?

You can apply Gram Schmidt in order to obtain decomposition of a matrix $A \in \Re^{n\times m}, n>m$ as:
\begin{align}
QR = A \quad Q \in \Re^{n \times n},R \in \Re^{n \times m}
\end{align}
where $Q$ is orthogonal matrix obtained by Gram Schmidt orthogonalisation and $R$ is right upper matrix with zero raws $r_i$ for $i > m$ . Espacially you can use it decomposition in order to solve minimalisation problem of the form:
\begin{align}
\min \limits_{x \in \Re^m \setminus \{0\}} ||Ax-b ||_2 ,\quad b \in \Re^n
\end{align}
