Question: Least squares and QR factorization. Suppose \(A\) is an \(m \times n\) matrix with linearly independent columns and QR factorization \(A=Q R\), and \(b\) is an \(m\) -vector. The vector \(A \hat{x}\) is the linear combination of the columns of \(A\) that is closest to the vector \(b\), i.e., it is the projection of \(b\) onto the set of linear combinations of the columns of \(A\).

1. Show that \(A \hat{x}=Q Q^{T} b\). (The matrix \(Q Q^{T}\) is called the projection matrix.)
2. Show that \(\|A \hat{x}-b\|^{2}=\|b\|^{2}-\left\|Q^{T} b\right\|^{2}\). (This is the square of the distance between \(b\) and the closest linear combination of the columns of \(A\).)

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