推 OckhamsRazor:推秒PO 06/02 14:49
課程名稱︰工程數學-線性代數
課程性質︰電機系必修
課程教師︰馮蟻剛
開課學院：電資學院
開課系所︰電機系
考試日期（年月日）︰2011年6月2日
考試時限（分鐘）：1:20~2:10 50分鐘
是否需發放獎勵金：是
（如未明確表示，則不予發放）
試題 :
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).)
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沒有伴隨著痛苦的教訓是沒意義的。
人如果不犧牲一些東西，就無法得到任何東西。
但是當超越了障礙，並且把得到的東西變成屬於自己的東西時...
人應該就能夠得到無法取代的鋼之心靈吧。
<Fullmetal Alchemist>
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