課程名稱︰線性代數 課程性質︰必修 課程教師︰馮蟻剛(不過其實四個老師考題都一樣) 開課學院：電資學院 開課系所︰電機系一年級 考試日期（年月日）︰2009 5/21 考試時限（分鐘）：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 : 1.Let A be the matrix defined by 2 1 1 A= 1 2 1 (註:這是方陣) 1 1 2 (a)(10%)Find the characteristic polynomial of A. (b)(20%)Find an invertible matrix P and a diagonal matrix D such that A=PDP^-1 (c)(10%)Does there exist a real matrix B such that A=B^2? If it does, find one such B.Is B unique?(You can express your answer in terms of P,P^-1,and D.) 2.Define the following vectors: 1 2 2 4 u1= 0 , u2= 1 , u3= -1 , v= 0 . -1 -1 -1 2 1 0 3 1 (註:上面是四個向量) (a)(20%)Find an orthogonal basis B={w1,w2,w3}for Span{u1,u2,u3}. (b)(10%)The vector v is in Span {u1,u2,u3}.Find c1,c2,c3 such that v=c1w1+c2w2+c3w3 (c)(10%)Find a vector w4 such that {w1,w2,w3,w4}forms an orthogonal basis for R^4. 3.(10%)Let the characteristic polynomials of the matrices A, B and C be respectively fA(t)=-t(t^2-1), fB(t)=-(t+1)(t^2+1), fC(t)=-(t+2)(t^2-1) Which matrix is not invertible? Which matrix is not diagonalizable? Which matrix is both invertible and diagonalizable? Explain your answer. (Answers without explanation get 0%.) 4.(10%)Let B1={v1,v2}and B2={v3,v4} be respectively a basis for W1 and W2 (subspaces of R^n). Prove that if the intersection of W1 and W2 is the zero subspace (i.e.,W1∩W2={0}),then {v1,v2,v3,v4} is linearly independent. --

※ 發信站: 批踢踢實業坊(ptt.cc) ◆ From: 140.112.249.177 ※ 編輯: princeeeeeee 來自: 140.112.249.177 (05/21 15:38) ※ 編輯: princeeeeeee 來自: 140.112.249.177 (05/21 15:38)