Gram-Schmidt Process
The problem, from first principles
Many useful properties (13-qr-decomposition's stable solving, an orthonormal basis's trivially-invertible-by-transpose structure) only apply when a set of vectors is not just independent, but orthonormal: mutually perpendicular, each of unit length. Most real vector sets you're handed aren't — Gram-Schmidt is the direct procedure for turning any independent set into an orthonormal one, one vector at a time, using nothing but 09-vector-projection's "subtract off the overlapping part" idea repeated against every earlier result.
From theory to code
Implement gram_schmidt(vectors), taking a list of linearly independent vectors and returning an orthonormal basis spanning the same space. The signature and docstring are already in the editor.
Constraints
vectorsis a list of 1D NumPy arrays, all the same length, guaranteed linearly independent.- Return a list of the same length, each entry a unit vector (
‖v‖ = 1), with every pair mutually orthogonal (v_i · v_j = 0fori ≠ j). - Process vectors in the given order — the resulting basis depends on that order (a different input order generally produces a different, still valid, orthonormal basis).
Hints
Hint 1
For each new vector, subtract its projection (09-vector-projection) onto every already-built basis vector, not just the most recent one — each earlier basis vector may capture a different direction of overlap.
Hint 2
Only normalize (divide by the norm) after all of that vector's projections onto earlier basis vectors have been subtracted — normalizing too early would rescale the vector before its remaining overlap has been fully removed.
Keywords
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Run your code to check it against the examples, or submit to run every test.