Gram-Schmidt Orthonormalization

Problem

Orthonormalize a list of vectors with the Gram-Schmidt process.

For each vector, subtract its projection onto every previously accepted basis vector, then normalize if the remaining norm exceeds 1e-12. Vectors that reduce to the zero vector are skipped. Return the orthonormal basis as a list of vectors.

Hint

A dependent vector subtracts away completely and should not be appended.

Test cases

case 1

in: [[[1,1],[1,0]]]

expected: [[0.7071067811865475,0.7071067811865475],[0.7071067811865477,-0.7071067811865474]]

case 2

in: [[[1,0],[0,1]]]

expected: [[1,0],[0,1]]

case 3

in: [[[1,1,0],[1,0,1]]]

expected: [[0.7071067811865475,0.7071067811865475,0],[0.40824829046386313,-0.40824829046386296,0.8164965809277261]]

case 4

in: [[[0,0],[1,1]]]

expected: [[0.7071067811865475,0.7071067811865475]]

case 5

in: [[[1,1],[2,2]]]

expected: [[0.7071067811865475,0.7071067811865475]]

Starter codePython

def gram_schmidt(vectors):
    # Your code here
    pass

5 test cases · run your code against them in the editor