課程名稱︰線性代數
課程性質︰選修
課程教師︰黃維信
開課學院：工學院
開課系所︰工程科學與海洋工程學系
考試日期（年月日）︰2009.4.24
考試時限（分鐘）：120
是否需發放獎勵金：是
（如未明確表示，則不予發放）
試題 :
┌2 3 3┐
1. Let A=│0 5 7│
└6 9 8┘
(a) Factor A into LU.(b) Find a basis for the row space of A.
(c) Find a basis for the column space of A.(15%)
2. True or false, with reason if true and counterexample if false:(25%)
(a) If columns 1 and 3 of B are the same, so are columns 1 and 3 of AB.
(b) If rows 1 and 3 of B are the same, so are rows 1 and 3 of AB.
2 2 2
(c) (AB)=A B
(d) If A and B are symmetric then AB is symmetric.
(e) If A and B are invertible then AB is invertible.
3. What is the relation between the rank r and the dimension of A when the
→ →
number of solutions to Ax = b is (10%)
→
(a) 0 or 1, depending on b.
→
(b) ∞, independing of b.
→
(c) 0 or ∞, depending on b.
→
(d) 1, regardless of b.
4
4. Find a basis for the following subspaces of R .(15%)
(a) The vectors for which X = 2X
1 4
(b) The vectors for which X +X +X = 0 and X +X = 0.
1 2 3 4 3
(c) The subspace spanned by (1, 1, 1, 1),(1, 2, 3, 4), and (2, 3, 4, 5).
→ T → T
5. Apply the Gram-Schmidt process to a = [0 0 1] , b = [0 1 1] ,
→ T
c = [1 1 1] , and write the result in the form A = QR.(15%)
6. Find the quadratic curve which gives the best least-squares fit to the
5
curve y = x between x = 0 and x = 1.(20%)
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