Add gram-schmidt implementation

Author: OliCheesman

linear_algebra/gram_schmidt.py

@@ -0,0 +1,56 @@
+import math
+
+
+def gram_schmidt(input_vectors: list[list[float]]) -> list[list[float]]:
+    """
+    Implements the Gram-Schmidt process to orthonormalize a set of vectors.
+    Reference: https://en.wikipedia.org/wiki/Gram%E2%80%93Schmidt_process
+
+    Case 1: Standard 2D Orthonormalization
+    >>> v1 = [[3.0, 1.0], [2.0, 2.0]]
+    >>> result1 = gram_schmidt(v1)
+    >>> [[round(x, 3) for x in v] for v in result1]
+    [[0.949, 0.316], [-0.316, 0.949]]
+
+    Case 2: 3D Vectors (The example from your error log)
+    >>> v2 = [[1.0, 1.0, 0.0], [1.0, 0.0, 1.0], [0.0, 1.0, 1.0]]
+    >>> result2 = gram_schmidt(v2)
+    >>> [[round(x, 3) for x in v] for v in result2]
+    [[0.707, 0.707, 0.0], [0.408, -0.408, 0.816], [-0.577, 0.577, 0.577]]
+
+    Case 3: Vectors that are already orthonormal (should remain unchanged)
+    >>> v3 = [[1.0, 0.0], [0.0, 1.0]]
+    >>> gram_schmidt(v3)
+    [[1.0, 0.0], [0.0, 1.0]]
+    """
+
+    orthonormal_basis: list[list[float]] = []
+
+    for v in input_vectors:
+        # Create a copy of the current vector to work on (equivalent to v_dash)
+        v_orthogonal = list(v)
+
+        for u in orthonormal_basis:
+            # Manual Dot Product: (v_dash * u)
+            dot_product = sum(vi * ui for vi, ui in zip(v_orthogonal, u))
+
+            # Manual Vector Subtraction & Scalar Mult: v_dash = v_dash - u * dot_product
+            v_orthogonal = [vi - (dot_product * ui) for vi, ui in zip(v_orthogonal, u)]
+
+        # Manual Norm Calculation: (v_dash * v_dash) ** 0.5
+        norm = math.sqrt(sum(xi**2 for xi in v_orthogonal))
+
+        if norm < 1e-15:
+            raise ValueError("The vectors are not linearly independent.")
+
+        # Manual Scalar Multiplication: u_new = v_dash * (1/norm)
+        u_new = [xi / norm for xi in v_orthogonal]
+        orthonormal_basis.append(u_new)
+
+    return orthonormal_basis
+
+
+if __name__ == "__main__":
+    import doctest
+
+    doctest.testmod()