課程名稱︰線性代數 課程性質︰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）︰2012/11/09 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） 試題 : 1. (24 points) True or false, with reason if true and counterexample if false: (a) For a linear system of three equations and four unknown variables, we should obtain multiple solutions. (b) The set of all invertible matrices M belongs to R nxn is a subspace of R nxn. (c) If A is a nonzero square matrix and A^3 is the zero matrix, then it is possible that A - I is singular. (d) A set V is a vector space if V satisfies the following properties: i. V has a zero vector ii. Whenever u and v belong to V, then u + v belongs to V. iii. Whenever v belongs to V and c is a scalar, then cv belongs to V. (e) If an n-element subset of a finite-dimensional vector space V is linearly independent, then the dimension of V is greater than n. (f) L(x,y) = (4x-y, 2x+3) is a linear transformation. 2. (12 points) Consider the following augmented matrix of a linear system in R: ┌ ┐ │ 1 β 1 │ 2 ｜ │ 1 2 2 │ 3 ｜ │ 1 3 β │ β+3 ｜ └ ┘ Determine all the possible value of β for the following cases: (a) This linear system has infinite solutions. (b) This linear system had no solution. (c) This linear system has a unique solution. 3. (10 points) Determine which of the following sets are subspaces of R^n? (a) A = {(x1,x2,...xn) ∈ R^n : x2 = 0 or xn = 0} (b) B = {(x1,x2,...xn) ∈ R^n : x1 + 3xn = 0} (c) C = {(x1,x2,...xn) ∈ R^n : sum(xn) = 1} (d) D = {(x1,x2,...xn) ∈ R^n : x1 = x2^2} (e) E = {(x1,x2,...xn) ∈ R^n : x3 is rational} ┌ ┐ │2｜ 4. (6 points) Suppose the complete solution to the equation Ax = ｜4｜ is ┌ ┐ ┌ ┐ ┌ ┐ ｜6｜ ｜2｜ ｜2｜ ｜0｜ └ ┘ x = ｜0｜+s｜0｜+t｜1｜. Find A. ｜0｜ ｜2｜ ｜0｜ └ ┘ └ ┘ └ ┘ ┌ ┐ ｜ 1 0 -3 1 2｜ 5. (8 points) Let A = ｜ 0 1 -4 -3 1｜ ｜-3 2 1 -8 -6｜ ｜ 2 -3 6 7 9｜ └ ┘ (a) Find the basis of R(A). (b) Find the basis of N(A). ┌ ┐ ┌ ┐ ｜a b｜ ｜ 0 d｜ 6. (10 points) Let V= F^2*2, W1 = {｜c a｜｜a,b,c ∈ F }, and W2 = {｜-d e｜｜ └ ┘ └ ┘ d,e ∈ F }. Find the dimensions of W1, W2, W1 + W2, and W1∩W2. 7. (10 points) Let W be the set of all vectors b = (b1, b2, b3) in R^3 such that the system of linear equations x1 - 2x2 - x3 = b1 2x1 - 3x2 + x3 = b2 5x1 - 8x2 + x3 = b3 has a solution. (a) Show that W is a subspace of R^3. (b) Find the dimention of W. 8. (10 points) Let a sequence of nxn matrices Ai be defined as ┌ ┐ ┌ ┐ ｜ 1 c1｜ ｜ 1 0 c1｜ A2 = ｜-c1 1｜ A3 = ｜ 0 1 c2｜ └ ┘ ｜-c1 -c2 1｜ └ ┘ ┌ ┐ ┌ ┐ │ 1 0 0 ．．． 0 c1 │ │ 1 0 0 c1│ │ 0 1 0 ．．． 0 c2 │ │ 0 1 0 c2│ │ 0 0 1 ．．． 0 c3 │ A4 =│ 0 0 1 c3│ Ai = │ ． ． ． ．．． ． ． │ │-c1 -c2 -c3 1│ │ 0 0 0 ．．． 1 c(i-1)│ └ ┘ │-c1 -c2 -c3 ．．． -c(i-1) 1 │ └ ┘ where c1 ∈ R, i = 1,2,...,n-1. Find the LU factorization of An. 9. (10 points) Suppose A, B ∈ R^m*n, a nonsingular Q ∈ R^m*m, and B = QA. Show that A and B have the same row space. -- ※ 編輯: danielu0601 來自: 140.112.249.18 (11/12 08:17)