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）︰2007/6/22
考試時限（分鐘）：
是否需發放獎勵金：是
（如未明確表示，則不予發放）
試題 :
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1.Apply the Gram-Schmidt process to a=[0 0 1] , b=[0 1 1],
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c=[1 1 1] and write the result in the form A=QR.(15%)
2.True or False, with reason if true and counterexample if false:(15%)
(a) If A and B are identical except that b = 2a ,then detB=2detA
11 11
(b) The determinant is the product of the pivots.
(c) If A is invertible and B is singular, then A+B is invertible.
(d) If A is invertible and B is singular, then AB is singular.
(e) The determinant of AB-BA is zero.
3. If P is an even permutation matrix and P is odd, deduce from P +P =
1 2 1 2
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P (P +P )P that det (P +P ) = 0. (10%)
1 1 2 2 1 2
4.Find the general solution to du/dt = Au if
┌0 -1 0┐
A=│1 0 -1│
└0 1 0┘
Can you find a time at which the solution u(T) is guaranteed to return to
the initial value u(0)? (20%)
5.True of false (with counterexample if false)(15%)
(a) If B is formed from A by exchanging two rows, then B is similat to A.
(b) If a triangular matrix is similar to a diagonol matrix, it is already
diagonol.
(c) If A and B are disgonolizable, so is AB.
6.Decide between a minimum, maximum, or saddle point fot the following
functions(20%)
(a) F=-1+4[e^(x)-x]-5xsiny+6y^2 at the point x=y=0.
(b) F=[x^(2)-2x]cosy,with stationary point at x=1,y=π.
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7.Compute A A and AA , and thier eigrnvalues and unit eigenvectors, for
┌1 1 0┐
A=│ │
└0 1 1┘
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Multiply the three matrices UΣV to recover A.(20%)
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