课程名称︰工程数学-线性代数 课程性质︰电机系必修 课程教师︰冯蚁刚 开课学院：电资学院 开课系所︰电机系 考试日期（年月日）︰2011年6月2日 考试时限（分钟）：1:20~2:10 50分钟 是否需发放奖励金：是 （如未明确表示，则不予发放） 试题 : Linear Algebra Quiz 2 Dept. of Elec. Eng.,National Taiwan University (Prof. Fong's Class) June 2 , 2011 USE OF ANT AUTOMATIC COMPUTING MACHINES IS PROHIBITED 1. Judge if the following statements are true of false. Give a concise proof to each true statement, and a counterexample to each false statement. (a) If A and B are n ×n matrices and v is an eigenvector of both A and B, then v is an eigenvector of AB. (20%) n ⊥ (b) For any subspace W of R , dim W = dim W. (20%) ┌ -4 0 2 ┐ 2. For the matrix M = │ 2 -2 -8 │find all eigenvalues and a basis for each └ 2 0 -4 ┘ eigenspace. (25%) Is the matrix M diagonalizable? Why or Why not? (5%) n 3. (a) Let P_w be the orthogonal projection for W, a subspace of R. Prove that P_w⊥ = I_n - P_w , where w⊥ is the orthogonal complement of W and I_n is the n ×n identity matrix. (10%) (b) Let W = Null B, where B is an m*n matrix of rank m. Prove that T T -1 P_w = I_n - B (B B ) B.(10%) ( Hint: For an n ×n matrix C with rank m and column space Col C, it is T -1 T known P_Col C = C (C C) C . Compare this formula to the one you are asked to prove and find the relation. ) ┌ -1 1 0 -1 ┐ (c) For A = │ 0 1 -2 1 │ and W = Null A, find the matrix P_w.(10%) │ -3 1 4 -5 │ └ 1 1 -4 3 ┘ ( Hint: You can apply the formula of (b), but note that rank A ≠ 4. You can also find a basis of W and use the Hint for(b).) -- 没有伴随着痛苦的教训是没意义的。 人如果不牺牲一些东西，就无法得到任何东西。 但是当超越了障碍，并且把得到的东西变成属於自己的东西时... 人应该就能够得到无法取代的钢之心灵吧。 <Fullmetal Alchemist> --

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