Let \(A=\begin{pmatrix}1&1\\1&2\\1&3\end{pmatrix}\) and \(b=(1,2,2)^\top\). Compute the least squares solution \(x^*\) and the residual \(\|Ax^*-b\|_2^2\).
Choose one option.
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B. \(x^*=(\tfrac{2}{3},\tfrac{1}{2})^\top\), residual \(\tfrac{1}{6}\)
The normal equations give \(x^*=(\tfrac{2}{3},\tfrac{1}{2})^\top\), and the residual \(Ax^*-b=(\tfrac{1}{6},-\tfrac{1}{3},\tfrac{1}{6})^\top\) has squared norm \(\tfrac{1}{6}\). The option with residual \(\tfrac{1}{3}\) uses the same \(x^*\) but adds the squared components incorrectly.
Optimization problem that minimizes the sum of squared residuals \(\|Ax-b\|_2^2\) to approximate a vector \(b\) by linear combinations of the columns of \(A\).