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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