課程名稱︰工程數學-線性代數
課程性質︰必修
課程教師︰馮世邁
開課學院：電機資訊學院
開課系所︰電機系
考試日期（年月日）︰2018/05/23
考試時限（分鐘）：09:20-10:10 （最後多延長5分鐘）
試題 :用右上角的*表示向量，及課本/上課的粗體
1. — —
｜ 1 -2 2 ｜
A = ｜ 8 11 -8 ｜
｜ 4 4 -1 ｜
— —
(a)(15%)Find the characteristic polynomial of A.
(b)(10%)Find all the eiganvalues of A and the multiplicity of each eiganvalue.
(c)(15%)Find a basis for each eiganspace of A.
(d)( 6%)Find and invertible matrix P and a diagonal matrix D such that
A=PDP^(-1)
(e)( 4%)Find the characteristic polynimail of A^(-1).
2. Let V = Span { u1* , u2* , u3* }, where
— — — — — —
｜ 1 ｜ ｜ 1 ｜ ｜ 3 ｜
u1*=｜-1 ｜ u2*=｜ 1 ｜ u3*=｜ 1 ｜
｜ 0 ｜ ｜ 1 ｜ ｜ 1 ｜
｜ 2 ｜ ｜ 3 ｜ ｜ 5 ｜
— — — — — —
(a)(20%)Find an orthogonal basis { v1*, v2*, v3* } for V.
(b)( 6%)Find an orthonormal basis {w1*, w2*, w3* } for V.
(c)( 9%)Let v* = [ 4 2 1 5 ]^(T) be a vector in V.
Express v* as a linear combanation of v1* v2* and v3*.
3.Let A be a 3x3 matrix with characteristic polynomial -t^3+p2t^2+p1t+p0
(a)(5%)Prove that A^T have the same characteristic polynomail as A.
(b)(5%)Find the characteristic polynomial of -A.
(c)(5%)Find the characteristic polynimail of A^2.