課程名稱︰線性代數一
課程性質︰必修
課程教師︰林紹雄
開課系所︰數學系
考試時間︰2006/1/7 14:00~17:00
試題 :
Math 201 14410-01 (Linear Algebra) Final Exam (1/7/2006)
There are Problems A to E with a total of 130 points. Please write down your
computational, or proof steps clearly on the answer sheets.
A. Let A be the matrix
[ 1 -1 0 ]
A = [ -1 1 -1 ]
[ 0 -1 1 ]
[ 0 0 -1 ]
4
Denote by V the column space of A in R . Each has 8 points in the following
problems.
(a) Use the Gram-Schmidt orthogonalization process to find an orthonormal
basis of V from the column vectors of A, and write down the QR-decomposition
of A.
(b) Perform the LDU-decomposition for A^T A to obtain the QR-decomposition
of A. Do you get the same answer as in (a)?
4
(c) Write down the projection matrix of the orthogonal projection from R
onto V, and find the minimal distance from the vector v=[1,-1,1,-1]^T to V.
(d) Find the best least square approximate solution to the linear system
x_1 - x_2 = 3, x_1 - x_2 + x_3 = a, x_2 - x_3 = -1, x_3 = 4
Where a is a constant. What condition should a satisfy so that this
approximate solution becomes an exact solution?
B. Let A be the matrix
[ -1 1 0 0 ]
A = [ -1 0 1 0 ]
[ 0 1 0 -1 ]
[ 0 0 -1 1 ]
Each has 6 points in the following problems.
(a) Draw a graph with numbered and directed edges (and numbered nodes) whose
incidence matrix is A.
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(b) Write down the Kirchhoff's Voltage Law to determine when a vector f 屬於 R
lies in the column space of A.
(c) How many independent loops exist in the graph?
(d) Is this graph a tree? Show that removing the last edge produces a spanning
tree.
C. (20 points) Find the characteristic polynomial, and the eigenvalues of the
matrix.
[ -1 0 -1 ]
A = [ 4 1 2 ]
[ 4 1 2 ]
Determine whether A is diagonalizable. If yes, find a matrix P such that
-1
P AP is a diagonal matrix.
D. Prove the following statements. Each has 10 points.
(a) Let A and B be m*n and n*m matrices respectively. Prove that
det [ 0 A ] = det(AB)
[ -B I_n ]
Moreover, if m＞n, prove that det(AB) = 0
n ┴ ┴ ┴
(b) Let V, W be two linear subspaces of R . Prove that (V+W) = V ∩ W
┴ ┴ ┴
and (V∩W) = V + W .
(c) If A is a nonsingular square matrix, prove that there exists a permutation
matrix P for which PA has no zero on its main diagonal.
E. Determine which of the following statements is true. Prove your answer ( or
give a counterexample). Each has 8 points.
3 2
(a) If a 2*2 matrix A satisfies A + 3A + 3A + I_2 = 0, then A must be
diagonalizable.
(b) Let P be a 3*3 permutation matrix so that det(P) = -1. Then det(P+I_3) = 0,
and there exists a family of invertible 3*3 matrices A(t) for 0 ≦t≦1
such that A(t) is continuous in t, and A(0) = -I_3, A(1) = P.
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(c) There exist three 6-dimensional linear subspaces U, V, W of R such that
U∩V∩W = {0}.
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※ 編輯: Nzing 來自: 140.112.240.198 (01/07 19:43)