課程名稱︰線性代數 課程性質︰系必修 課程教師︰馮世邁 開課學院：電資學院 開課系所︰電機系 考試日期（年月日）︰99年6月3日 考試時限（分鐘）：50 min 是否需發放獎勵金：是!感謝! 試題 : 1.Let A and u be defined below. Let W=Col A. ┌ ┐ ┌ ┐ │ 1 1 1 1 │ │ 1 │ A = │ 0 1 -1 -1 │ , u= │ 0 │ │ 1 1 1 1 │ │-1 │ │ 0 1 -1 2 │ │ 2 │ └ ┘ └ ┘ (a) (20%) Find an orthonormal basis Β fow W. (b) (10%) Extend Β to form an orthonormal basis for R^4. (c) (5%) Find P_W. (d) (5%) Find the orthogonal projection of u onto W. ┴ (e) (5%) Find the orthogonal projection of u onto W . 2. ┌ ┐ │ 4 2 2 │ A = │ 2 4 2 │ │ 2 2 4 │ └ ┘ (a) (25%) Find orthogonal P and diagonal D such that A=PDP^T (b) (5%) Find the spectral decomposition of A.(You may express your answer in the form of u_i u_i^T.) (c) (5%) Is A invertible? If it is, find the spectral decomposition of A^-1. 3. (10%) Show that 2 eigenvectors corresponding to distinct eigenvalues of a symmetric matrix are orthogonal. 4.(10%) Let S_1 ={u_1, u_2, ......,u_k } be a linearly independent subset of R^n and let u be a linear combination of vectors in S_1. Let S_2 ={u_1, u_2, ......,u_k, u}. Suppose we apply the Gram-Schimdt process to S_1 and S_2 to obtain the following orthogonal sets: S'_1 ={v_1, v_2, ......, v_k}, S'_2 ={v_1, v_2, ......, v_k, v}. What is v? (Explain your answer) --

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