題組內容

2. Let A be an $m \times n$ matrix with independent columns, and let A = QR be its QR factorization. The orthogonal projection of any vector $\underline{b} \in R^m$ onto the column space of A can be represented by $\underline{x} = P\underline{b}$, where P is the projection matrix and $\underline{x} \in \text{Col}(A)$ as shown in Figure 1.

(b) (5%) Let \( Q = \left[ \underline{q}_1 \quad \underline{q}_2 \quad \dots \quad \underline{q}_n \right] \). Show that \( \underline{x} \) can be written as a linear combination of \( \underline{q}_1, \underline{q}_2, \dots, \underline{q}_n \) and represented by
$\underline{x} = (\underline{q}_1^T\underline{b})\underline{q}_1 + (\underline{q}_2^T\underline{b})\underline{q}_2 + \dots + (\underline{q}_n^T\underline{b})\underline{q}_n$