Gram-Schmidt Orthonormalizationeasy

Gram-Schmidt Orthonormalization

easy Math for ML 15-20 min

NumPy allowed · plain Python earns 10 bonus points

A vector transforms

Matrix multiplication is the first computation behind every model.

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01vector

02matrix

03transform

04result

[2, −1] × [[1, 2], [−.5, 1]] = [0, −2]

Computed matrix × vector transformation

Construct an orthonormal basis using the classical Gram-Schmidt orthogonalization process.

Related Skills

Gram-Schmidt Process

Task

Implement gram_schmidt(vectors): sequentially project each vector onto the span of previously computed basis vectors, subtract projections to get orthogonal vector w, normalize w to unit length, and append to basis. Discard vectors whose remaining norm is < 1e-7. Round coordinates to 4 decimals.

Function Signature

gram_schmidt(vectors: list[list[float]]) -> list[list[float]]

Examples

Example 1: 2D Standard Basis
Input: vectors = [[1, 1], [0, 1]]
Output: [[0.7071, 0.7071], [-0.7071, 0.7071]]
Explanation: Orthogonalizing 2D plane vectors.

Example 2: Already Orthogonal
Input: vectors = [[2, 0], [0, 3]]
Output: [[1, 0], [0, 1]]
Explanation: Normalizes already orthogonal inputs.

Constraints

1 <= len(vectors) <= 10.
Dimension 1 <= d <= 10.

Python3Saved ✓

def gram_schmidt(vectors: list[list[float]]) -> list[list[float]]:

"""Produce an orthonormal basis from linearly independent vectors using Gram-Schmidt.

Discard vectors with norm < 1e-7. Round coordinates to 4 decimals."""

# Your implementation here

pass

Case 1: 2D Standard Basis

Input: vectors = [[1, 1], [0, 1]]

Expected: [[0.7071, 0.7071], [-0.7071, 0.7071]]

Case 2: Already Orthogonal

Input: vectors = [[2, 0], [0, 3]]

Expected: [[1, 0], [0, 1]]