課程名稱︰線性代數
課程性質︰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）︰2012/05/24
考試時限（分鐘）：50
是否需發放獎勵金：是
（如未明確表示，則不予發放）
試題 :
1.Judge if the following statements are true or false. Give a concise proof to
each true statement, and a counterexample to each false statement.
(a)If an m by n matrix satisfies A^k = O, the zero matrix,for some positive
integer k ,then 0 is the only eigenvalue of A. 15%
2 0 -1
(b)The matrix -1 3 -1 is diagonalizable. 15%
2 0 5
(c)If the m by n matrices A and B are diagonalizble then A+B is
diagonalizable. 15%
2. Let u1 = [1 -1 0 1 1]^T, u2 = [2 -1 0 3 2]^T, u3 = [1 -1 1 1 1]^T and
u4 = u2 + 2u3 - u1. Apply the Gram-Schmidt process to these vectors
sequentially to find a set of four orthogonal vectors. 25%
3.(a) Let B = [b1 b2 ...bn] be a invertible n by n matrix and C = [c1 c2 ..cn].
Show that the matrix transformation T indeced by C(B^-1) statisfies
T(bj) = cj for j = 1,2,3.....n. 15%
(b) Let a linear operator U from R^n ti R^n be defined by U(bj) = bj+1
(注:j+1是下標) for j = 1,2,3...n-1 and U(bn) = 0. Prove that U^n = O,
the zero operator, where U^n = UxUxUxU.....(n times). 15%
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