Three measurements are collected:
A line y^=w0+w1x\hat{y} = w_0 + w_1 xy^=w0+w1x is fitted by solving the normal equations
w=(X⊤X)−1X⊤yw = (X^\top X)^{-1}X^\top yw=(X⊤X)−1X⊤y
where XXX carries a leading column of ones, so XXX is 3×23 \times 23×2.
Define the residual vector as r=y−Xwr = y - Xwr=y−Xw. What are rrr and X⊤rX^\top rX⊤r?
Select all that apply.
r=(−1, 2, −1)⊤r = (-1,\ 2,\ -1)^\topr=(−1, 2, −1)⊤ and X⊤r=(0, 0)⊤X^\top r = (0,\ 0)^\topX⊤r=(0, 0)⊤
r=(1, −2, 1)⊤r = (1,\ -2,\ 1)^\topr=(1, −2, 1)⊤ and X⊤r=(0, 0)⊤X^\top r = (0,\ 0)^\topX⊤r=(0, 0)⊤
r=(−17, −167, 117)⊤r = \left(-\tfrac{1}{7},\ -\tfrac{16}{7},\ \tfrac{11}{7}\right)^\topr=(−71, −716, 711)⊤ and X⊤r=(−67, 0)⊤X^\top r = \left(-\tfrac{6}{7},\ 0\right)^\topX⊤r=(−76, 0)⊤
r=(1, −2, 1)⊤r = (1,\ -2,\ 1)^\topr=(1, −2, 1)⊤ and X⊤r=(0, 6)⊤X^\top r = (0,\ 6)^\topX⊤r=(0, 6)⊤