課程名稱︰管理數學 課程性質︰必修 課程教師︰謝承熹 開課學院：管理學院 開課系所︰財金系 考試日期（年月日）︰98/01/12 考試時限（分鐘）：160分鐘 是否需發放獎勵金：是 （如未明確表示，則不予發放） 試題 : Ⅰ.(40 points) Answer each of the following as true (T) or false (F). Justify your answer, or you cannot get any points. Moreover, please give your answers in order. Thanks! 1.Let A be an m×n matrix. If B is a nonsingular n×n matrix, then AB and A have the same null space. 2.Let A, B, S, and T are n×n matrices. Suppose that A=ST and B=TS, where S is nonsingular, then A is similar to B. 3.Suppose an n×n matrix A is positive definite. Let B be an n×m matrix with rank m, then B'AB is positive definite. 4.Let A be a diagonalizable, n×n idempotent matrix, then each eigenvalue of A is either 0 or 1. 5.(Continue) You are given the result that rank B = rank(PBQ) for nonsingular matrices P and Q. If 0 is an eigenvalue of A with multiplicity k, k ≦ n, then rank A = k. Ⅱ.(15 points) Find λ≠-0.5 such that v1, v2, and v3 is linear dependent, ╭ ╮ ╭ ╮ ╭ ╮ │ λ │ │-0.5│ │-0.5│ where v1=│-0.5│, v2=│ λ │, v3=│-0.5│. │-0.5│ │-0.5│ │ λ │ ╰ ╯ ╰ ╯ ╰ ╯ Ⅲ.(15 points) If the rank of the set of vectors {b1, b2, b3} is equal to that of the set of vectors {a1, a2, a3}, and b3 can be representation as the linear combination of {a1, a2, a3}, find the value of a and b, where ╭ ╮ ╭ ╮ ╭ ╮ ╭ ╮ ╭ ╮ ╭ ╮ │ 0│ │ a│ │b│ │ 1│ │3│ │ 9│ b1=│ 1│, b2=│ 2│, b3=│1│, a1=│ 2│, a2=│0│, a3=│ 6│. │-1│ │-1│ │0│ │-3│ │1│ │-7│ ╰ ╯ ╰ ╯ ╰ ╯ ╰ ╯ ╰ ╯ ╰ ╯ Ⅳ.(15 points) Prove that for any matrix A, rank A = rank(A'A).(Hint: Prove that the null space of A and A'A is the same.) ╭ ╮ ╭ ╮ │3│ │a│ Ⅴ.(15 points) Consinder the vector v=│2│ in R^3. Let W={│b│}. Find the │6│ │b│ ╰ ╯ ╰ ╯ ╭ ╮ ╭ ╮ ╭ ╮ │a│ │1│ │0│ projection of v onto W.(Hint: Note that │b│=a│0│+b│1│.) │b│ │0│ │1│ ╰ ╯ ╰ ╯ ╰ ╯ --

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