Solve normal equations

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Problem

Fit $\hat{y}=b+wx$ by ordinary least squares to $(0,1)$ , $(1,2)$ , and $(2,2)$ . Form $X^{\mathsf T}X$ and $X^{\mathsf T}y$ using an intercept column, then solve for $(b,w)$ .

Reveal answer or reference solution

The normal-equation quantities are

X^{\mathsf T}X=\begin{bmatrix}3&3\\3&5\end{bmatrix},\qquad X^{\mathsf T}y=\begin{bmatrix}5\\6\end{bmatrix}.

Solving gives $b=\frac{7}{6}$ and $w=\frac{1}{2}$ . The fitted values are $(\frac{7}{6},\frac{5}{3},\frac{13}{6})$ .

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Solve normal equations

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