補充
※ 引述《syuusyou (syuusyou)》之銘言：
: 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.
第三題與第五題亦同，正確須解釋，錯誤舉反例。
: (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.
是請說明，否請舉反例。
: (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)
前兩小題之 T 形如：
x y z
┌ ┐
x│ │
y│ │
z│ │
└ ┘
x,y,z為空間中右手系坐標。
: (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)
前三小題答案形式為：
span { vector, vector, ... }
若僅列出basis但未註明span會酌予扣分。(考完聽助教講的)
: ┌ ┐
: │ 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?
是請說明，否請舉反例。
: (a) T: R^(nxn) -> R by T(A) = tr(A)
tr(A)：trace of A，矩陣A主對角線上元素和
: (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)
Ri[x]：i次多項式函數
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