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 $.

阿摩線上測驗 · 2026-10-22T09:06:53+08:00

3. Let L: R^2 → R^2 denote the mapping of a vector x onto its orthogonal projection on a line $l$ passing through the origin, which forms an angle θ with the $X$-axis as shown in Figure 2. Let $A$ be the matrix that corresponds to this mapping, i.e., L(x) = Ax. Find $A$ by each method below. (a) (10%) Method 1: According to the formula A = [L(e_1)L(e_2)], where e_1 = bmatrix 1 \ 0 bmatrix and e_2 = bmatrix 0 \ 1 bmatrix, A is obtained by evaluating the mappings on the standard basis vectors e_1 and e_2.

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