Past exam of the mathematics course of the University of Cambridge / 2020 / ib / Paper 2 / 18H / a

Solution

Because

X

X^{T} X

β = (X^{T} X)^{- 1} X^{T} Y .

It satisfies the normal equations

X^{T} (Y - X β) = 0,

so the residual is orthogonal to the column space of

X

. For any

β

Y - Xβ = (Y - X β) + X (β - β) .

S (β) = S (β) + (β - β)^{T} X^{T} X (β - β) \geq S (β) .

Thus $β$ minimizes the least-squares objective.

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