推 lucas12417 :已收精華區 01/19 23:08
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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