推 liberty5566 :才考完一個小時...........你手腳也太快了@@ 11/09 12:54
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※ 編輯: danielu0601 來自: 140.112.30.132 (11/09 13:43)
推 cchao28 :∩ 11/09 17:32
推 Parazicecum :∩ 11/09 18:30
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-dimentional 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 交集(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.
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