Gram-Schmidt Next Orthonormal Vector

Problem

Extend an orthonormal pair q1, q2 to include vector a by one step of Gram-Schmidt.

Subtract the projections of a onto q1 and q2, normalize the residual, and return the resulting unit vector. Return None if a lies in span(q1, q2), meaning the residual norm is below 1e-12.

Hint

Assume q1 and q2 are already orthonormal; only one vector needs to be produced.

Test cases

case 1

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

expected: [0,0,1]

case 2

in: [[1,0,0],[0,1,0],[2,0,0]]

expected: null

case 3

in: [[1,0,0],[0,1,0],[0,3,4]]

expected: [0,0,1]

case 4

in: [[0.7071067811865476,0.7071067811865476,0],[0,0,1],[1,1,1]]

expected: null

case 5

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

expected: null

Starter codePython

def gram_schmidt_next(q1, q2, a):
    # Your code here
    pass

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