[試題] 99上 李明穗 線性代數 期末考
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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※ 編輯: arthur104 來自: 123.193.37.84 (01/13 23:18)
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