(a) (10%) Show that P = QQ^T

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.

3. Let $L: R^2 \to R^2$ denote the mapping of a vector $\underline{x}$ onto its orthogonal projection on a line l passing through the origin, which forms an angle $\theta$ with the X-axis as shown in Figure 2. Let A be the matrix that corresponds to this mapping, i.e., $L(\underline{x}) = A\underline{x}$. Find A by each method below.(a) (10%) Method 1: According to the formula $ A = \left[L\left(\underline{e}_1\right)L\left(\underline{e}_2\right)\right] $, where $ \underline{e}_1 = \begin{bmatrix} 1 \\ 0 \end{bmatrix} $ and $ \underline{e}_2 = \begin{bmatrix} 0 \\ 1 \end{bmatrix} $, $ A $ is obtained by evaluating the mappings on the standard basis vectors $ \underline{e}_1 $ and $ \underline{e}_2 $.

題組內容

(a) \( (10\%) \) Show that \( P = QQ^{T} \)

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