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︰電機工程學系
考試日期（年月日）︰2018/04/25
考試時限（分鐘）：10:20-12:10 （最後好像延長10分鐘）
試題：
用右上角的*，表示向量(即課本/上課的粗體)
Use of all automatic computing machines including calculator is prohibited.
1. — —
｜ 1 0 3 1 ｜
A = ｜ 2 -1 5 1 ｜
｜-1 1 -2 1 ｜
｜ 0 1 1 1 ｜
— —
(a)(15%)Find an invetible matrix P and the reduced row echelon form R such
that PA=R.
(b)(5%) Find another invertible matrix P' such that P'A=R.
(Notice that P'≠P)
2.(4+8%) Define T：R^3→R^3 by
— —
x1 ｜ 4x1+ x2-x3 ｜
T ( [ x2 ] ) = ｜ - x1- x2 ｜
x3 ｜ -5x1-3x2+x3 ｜
— —
Find the standard matrix of T and the inverse of T.
3. Let U1：R^n→R^m and U2：R^m→R^p be linear.
(a)(5%)Prove that if U1 is not one-to-one, then U2U1 is not one-to-one.
(b)(5%)If U2U1 is onto, can we say that U2 is onto? (Justify your answer.)
4. — —
｜ 1 -1 2 1 ｜
｜ 2 -1 - 1 4 ｜
A = [a1* a2* a3* a4*] = ｜-4 5 -10 -6 ｜
｜ 3 -2 10 -1 ｜
— —
(a)(12%)Find detA.
(b)(8%) Find the determinant of the following matrices：
(i) -A
(ii) [a4* a3* a2* a1*]
(iii)[a1* a2* a3* 2a4*-3a2*]
(iv) [2a1*-a2* a2*+3a3* 7a3*-4a1* a1*+a2*+a3*].
5. — —
｜-1 2 1 -1｜
A = ｜ 2 -4 -3 0｜
｜ 1 -2 0 3｜
— —
(a)(16%)Find a basis for each of the following：
(i) Col A
(ii) Null A
(iii) Row A
(iv) Col A^T
(b)(10%)Let B be the basis of Null A that you obtained in Part (a). Find a
basis for R^4 that contains B.
6.
(a)(6%)Let A1, A2 be m x n matrices. Prove that V = {v ∈ R^n：A1v*=A2v*} is
a subspace of R^n.
(b)(6%)Let B be an n x (n-1) matrix with rank B = n-1. Prove that
W = {w* ∈ R^n：[B w*] is not invertible} is a subspace of R^n.