The Gram-Schmidt process converts a set of linearly independent vectors into an orthogonal (or orthonormal) set that spans the same subspace.

Algorithm:

Starting with vectors $\mathbf{v}_1, \mathbf{v}_2, \ldots, \mathbf{v}_n$:

• $\mathbf{u}_1 = \mathbf{v}_1$
• $\mathbf{u}_k = \mathbf{v}_k - \sum_{j=1}^{k-1} \frac{\mathbf{v}_k \cdot \mathbf{u}_j}{\mathbf{u}_j \cdot \mathbf{u}_j} \mathbf{u}_j$ (orthogonal vectors)
• $\mathbf{e}_k = \frac{\mathbf{u}_k}{\|\mathbf{u}_k\|}$ (orthonormal vectors)

Applications:

• QR decomposition: $A = QR$ where $Q$ has orthonormal columns
• Solving least squares problems
• Finding orthonormal bases for subspaces