Normal Equations 2x2

Problem

Solve the linear least squares problem for a design matrix A (n x 2) and response b (length n) via the normal equations:

(A^T A) x = A^T b

Return the coefficients [x0, x1], or None if A^T A is singular.

Hint

Build the 2x2 Gram matrix A^T A and the right-hand side A^T b, then solve.

case 1

in: [[[1,1],[1,2],[1,3]],[1,2,3]]

expected: [0,1]

case 2

in: [[[1,1],[1,2],[1,3]],[1,3,4]]

expected: [-0.3333333333333333,1.5]

case 3

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

expected: [0,1]

case 4

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

expected: null

Starter codePython

def normal_equations(A, b):
    # Your code here
    pass

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