[試題] 99上 李明穗 線性代數 期中考

[syuusyou (syuusyou)]

課程名稱︰線性代數
課程性質︰系必修
課程教師︰李明穗
開課學院：電機資訊學院
開課系所︰資訊系
考試日期（年月日）︰2010/11/11
考試時限（分鐘）：150
是否需發放獎勵金：是
（如未明確表示，則不予發放）

Problem 1 (24 points)

(True of False) Please provide brief explanation if the answer is positive. If
the answer is negative, please give a counterexample. Otherwise, no credits
will be given.

    (a) If A and B are square matrices, and AB is invertible, then A and B are
        invertible.

    (b) If A is an nxn matrix and the system Ax = 0 has nontrivial solutions,
        then A is not invertible.

    (c) If AB = -BA, the nat least one of A, B is not invertible.

    (d) Let A, B be nxn matrices such that AB = I, then BA = I.

    (e) If A is an mxn matrix and n < m, then the equation Ax = 0 will have
        infinite many answers.

    (f) If the reduced row echelon form of [A b] contains a zero row, then
        Ax = b has infinitely many solutions.

    (g) Let A be an mxn matrix. Then the rank of A is n if and only if the
        equation Ax = b has at most one solution for each b in R^m.

    (h) If a system of linear equation in R has two different solutions, then
        it must have infinitely many solutions.

Problem 2 (8 points)

        ┌         ┐
        │ 1  0  1 │
        │ 0  1  2 │
Let A = │ 2 -1  0 │
        │ 1 -1 -1 │
        └         ┘

    (a) Please find a basis of all x's that satisfy Ax = 0.

    (b) Find a basis of all b's for which Ax = b has feasible solutions.

Problem 3 (15 points)

Determine whether or not the following are subspace of R^2.

    (a) {(x1, x2) | x1 + x2 = 0}

    (b) {(x1, x2) | x1*x2 = 0}

    (c) {(x1, x2) | x1 = 3*x2}

    (d) {(x1, x2) | x1 = |x2|}

    (e) {(x1, x2) | x1 = 3*x2 + 1}

Problem 4 (9 points)

                          ┌  ┐ ┌  ┐   ┌  ┐ ┌  ┐   ┌  ┐ ┌  ┐
                          │ 3│ │ 2│   │ 2│ │-1│   │-1│ │ 5│
        T: R^3 -> R^3 by T│-2│=│ 1│, T│ 1│=│ 3│, T│ 3│=│-1│;
                          │ 1│ │-3│   │-3│ │ 2│   │ 2│ │10│
                          └  ┘ └  ┘   └  ┘ └  ┘   └  ┘ └  ┘
                 ┌  ┐
                 │ 5│
        What is T│-1│?
                 │10│
                 └  ┘

Problem 5 (12 points)

    (a) Let T: R^3 -> R^3 be the transformation which rotate 60 degrees along
        z-axis and then reflect through the xy-plane. Please write down T.

    (b) Let T': R^3 -> R^3 be the transformation which reflect throught the
        xy-plane and then rotate 60 degrees along z-axis. Please write down
        T'.

    (c) Let A belongs to R^n, B belongs to R^m, and T is the linear
        transformation from A to B. Can we always find T' which works as the
        inverse transform from B back to A? (Please state your reason)

Problem 6 (20 points)


                ┌              ┐
                │ 1  4  5  6  9│
        Let A = │ 3 -2  1  4  1│
                │-1  0 -1 -2 -1│
                │ 2  3  5  7  8│
                └              ┘

    (a) Find the row space of A.

    (b) Find the column space of A.

    (c) Find the null space of A.

    (d) Find rank(A) and nullity(A).

Problem 7 (12 points)

Which of the followings are linear?

    (a) T: R^(nxn) -> R by T(A) = tr(A)

    (b) T: R^(nxn) -> R by T(A) = det(A)

    (c) T:R^3 -> R^2 by T(a, b, c) = (a-b, 2c)

    (d) T: R2[x] -> R3[x] by T(f(x)) = xf(x) + f'(x)

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