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）︰2011/1/13 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 : Problem 1 (18 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 u is orthogonal to v+w,then u is orthogonal to v and w. (b) Every orthonormal set in R^n is linearly independent. (c) Let A be an nxn real symmetric matrix. For any column vectors x,y in R^n, define <x,y>=y^T Ax. Then <,> is an inner product on R^n. (d) Assume A is a mxn matrix,m≧n,and B is a mxp matrix. If X is an nxp unknown matrix,then the system A^T AX = A^T B always has a solution. (e) For an inconsistent linear system Ax=b, we can find its least-squares solution (A^T A)^(-1) A^T b. However,if (A^T A) is not invertible, there is no least-squares solution. (f) Let A be an mxn matrix and the system of linear equations Ax=b be inconsistent. Then the vector z in R^n that minimizes ||Ax-b|| is unique. (g) Let W1 and W2 be subspaces in R^3 with equations x-y+2z=0 and x-y-z=0, respectively, then W1 and W2 are orthogonal complements. (h) Let S1 and S2 be subsets if R^n and S1┴ = S2┴, then S1=S2. (i) det(cA)=c*det(A), where c is a scalar. Problem 2 (16 points) ╭ 1 2 3 ╮ Given a matrix A │-1 0 -3 │. ╰ 0 -2 3 ╯ (a) (12 points) Use the Gram-Schmidt process to factor A into a product QR where Q = (q1,q2,q3) is a matrix with orthonormal column vectors {q1,q2,q3} and R is an upper triangular matrix with positive diagonal entries. (b) (4 points) If x = 2q1 + 2q2 + q3 and y = 5q1 + q3,determine the vector norm ||x|| and the inner product <x,y>. Problem 3 (8 points) ╭ ╮ ╭ ╮ T -1 -1 (a) Let A=│ 4 2 │, B=│ 2 1 │,find det(3A A BAB ). │11 6 │ │ 7 3 │ ╰ ╯ ╰ ╯ ╭ ╮ (b) Find det │2I B │= _______ │ 0 AB │ ╰ ╯ Problem 4 (8 points) ╭3 7 2 9 5╮ │0 2 0 7 1│ If λ1,λ2,λ3,λ4,λ5 are all the eigenvalues of the matrix:│8 5 2 7 9│ │7 3 5 2 0│ 2 2 2 2 2 ╰3 0 9 0 2╯ then what is the value of λ1 +λ2 +λ3 +λ4 +λ5 ? Problem 5 (18 points) ╭ 5 3 -7 ╮ Let A = │-1 1 1 │, ╰ 3 3 -5 ╯ (a)(12 points) Find the eigenvalues and the corresponding eigenvectors. (b)( 3 points) Diagonalize matrix A. n (c)( 3 points) Finf A , where n is any positive integer. Problem 6 (16 points) ╭ a b c ╮ Students A and B were solving the eigenvalues of the matrix,M =│ 0 d 1 │. ╰ 0 2 e ╯ However, student A mistook the value of d and obtained the eigenvalues 0,1,and 3. Student B mistook the value of e and obtained the eigenvalues 1,1,and -2. Assume no other mistakes happened during their computation (a)(3 points) Find the value of a. (b)(5 points) If the sum of correct eigenvalues of M is 1. Find the correct values of d and e. (c)(5 points) Find the correct eigenvalues of M. Problem 7 (8 points) ╭ a b c ╮ ╭ 4a+5d+6g 4b+5e+6h 4c+5f+6i ╮ Given A = │ d e f │,det(A) = 5, B = │ 2a+3d 2b+3e 2c+3f │, ╰ g h i ╯ ╰ a b c ╯ find det(B). Problem 8 (10 points) ┌-1 2 ┐ ┌4┐ Let A = │ 2 -3 │and b = │1│ └-1 3 ┘ └2┘ (a)(5 points) Please find x belonging to R^2, so that ||Ax-b|| is minimized. (b)(5 points) Find the orthogonal projection of b onto the column space of A. --

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