The Gram–Schmidt process in \( \mathbb{R}^n \)

Let \( \{v_1,\dots,v_k\} \) be a family of vectors in an inner product space, for example \( \mathbb{R}^n \) with the standard dot product. The Gram–Schmidt process constructs an orthogonal family \( \{u_1,\dots,u_k\} \) that spans the same subspace, and, after normalisation, an orthonormal family \( \{e_1,\dots,e_k\} \).

If for some \( j \) the vector \( u_j \) is the zero vector, then \( v_j \) lies in the span of \( \{v_1,\dots,v_{j-1}\} \) and does not contribute a new direction to the subspace. In this case the orthonormal basis has fewer vectors than the original family.

Geometric interpretation

At each step you subtract from \( v_j \) its projections onto the previously constructed orthogonal vectors. Geometrically, you are removing from \( v_j \) all components along the known directions, leaving only the component in the new independent direction. In the standard Euclidean setting:

Gram–Schmidt repeatedly subtracts such projections so that each new \( u_j \) is orthogonal to the subspace spanned by the earlier vectors.

Worked example in \( \mathbb{R}^3 \)

Consider the vectors

1. First vector: \[ u_1 = v_1 = (1,1,0), \quad \|u_1\| = \sqrt{1^2 + 1^2} = \sqrt{2}, \quad e_1 = \frac{1}{\sqrt{2}}(1,1,0). \]
2. Second vector: \[ u_2 = v_2 - \frac{\langle v_2,u_1\rangle}{\langle u_1,u_1\rangle} u_1. \] We have \( \langle v_2,u_1\rangle = 1\cdot1 + 0\cdot1 + 1\cdot0 = 1 \) and \( \langle u_1,u_1\rangle = 2 \), so \[ u_2 = (1,0,1) - \frac{1}{2}(1,1,0) = \left(\tfrac{1}{2}, -\tfrac{1}{2}, 1\right). \] Then \[ \|u_2\| = \sqrt{\tfrac{1}{4} + \tfrac{1}{4} + 1} = \sqrt{\tfrac{3}{2}}, \quad e_2 = \frac{u_2}{\|u_2\|}. \]
3. Third vector:

   Subtract the projections onto both \( u_1 \) and \( u_2 \). The calculator automates these arithmetic steps and shows the resulting orthogonal and orthonormal vectors.

You can reproduce this example in the calculator by setting \( n = 3 \), 3 vectors, and entering the components column by column.

Numerical aspects and stability

• Classical vs. modified Gram–Schmidt: This tool implements the classical algorithm. In floating-point arithmetic, modified Gram–Schmidt or QR factorisation with Householder reflections often behaves better numerically, especially when vectors are nearly linearly dependent.
• Linear dependence detection: When the squared norm \( \|u_j\|^2 \) falls below a small threshold, the vector is treated as zero and marked as linearly dependent.
• Scaling: Very large or very small components can amplify round-off errors. Rescaling vectors (for example dividing by a common factor) before applying Gram–Schmidt often improves stability.

Frequently asked questions

Do the orthogonal and orthonormal families span the same subspace as the original vectors?

Yes. At each step \( u_j \) is obtained from \( v_j \) by subtracting a combination of \( u_1,\dots,u_{j-1} \), which themselves lie in the span of \( v_1,\dots,v_j \). This guarantees that \( \mathrm{span}\{u_1,\dots,u_j\} = \mathrm{span}\{v_1,\dots,v_j\} \) for all \( j \).

What happens if I input more than \( n \) vectors in \( \mathbb{R}^n \)?

Any family of more than \( n \) vectors in \( \mathbb{R}^n \) is necessarily linearly dependent. The calculator will produce at most \( n \) non-zero orthogonal vectors; the remaining ones will appear as zero vectors and be flagged as dependent.

Why are some entries very small numbers instead of exact zeros?

Because the implementation uses floating-point arithmetic, operations like subtraction and projection can introduce small numerical errors. Values extremely close to zero (for example \( 10^{-12} \)) should be interpreted as numerical zeros; most exercises round these values to 0.

Can I use the result as an orthonormal basis for QR factorisation?

Conceptually, yes: the columns of the orthonormal matrix \( Q \) in a QR factorisation can be obtained via Gram–Schmidt. However, for serious numerical work it is better to rely on a QR algorithm designed for stability (Householder reflections, Givens rotations) rather than hand-rolled Gram–Schmidt.