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 : T T 1.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%) 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 T T 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. T T 7.Compute A A and AA , and thier eigrnvalues and unit eigenvectors, for ┌1 1 0┐ A=│ │ └0 1 1┘ T Multiply the three matrices UΣV to recover A.(20%) --

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