課程名稱︰線性代數
課程性質︰系必修
課程教師︰陳文進
開課學院：電資學院
開課系所︰資訊系
考試日期（年月日）︰2009/11/12
考試時限（分鐘）：210 min.(14:30 ~ 18:00)
是否需發放獎勵金：是
（如未明確表示，則不予發放）
試題 :
Fall 2009 線性代數期中考 Exam time: 2:30 ~ 6:00
┌ ┐
│ 1 1 2 2 3 │
1. (10%) What is the rank of the matrix A =│ 0 1 1 -1 -1 │
│ 2 3 a+2 3 a+6│
│4 0 4 a+7 a+11│
└ ┘
2. (10%) If A is an mxn matrix. Prove that the equations Ax = 0 and (A^T)Ax=0
have the same solutions. (A^T is the transpose of A).
3. (10%) If A, B are both invertible nxn matrix, explain briefly if each of the
following five statements is true or false.
a. (kA)^(-1) = k(A^(-1)) (k != 0)
b. (A^2)^(-1) = (A^(-1))^2
c. (A+B)^(-1) = A^(-1) + B^(-1)
d. (A^(-1) + B^(-1))^(-1) = A + B
e. (AB)^(-1) = A^(-1)B^(-1)
┌ ┐
│1 1 2│
4. (10%) Find the LU - decomposition of the matrix A =│2 1 3│
│3 3 7│
└ ┘
┌ ┐
│1 2 1 2│
5. (10%) Let A =│0 1 a a│,
│1 a 0 1│
└ ┘
Find the value of a such that dim{xεR^4 | Ax = 0} = 2.
6. (10%) Find the basis for the subspace
┌ ┐ ┌ ┐ ┌ ┐ ┌ ┐
│1│ │0│ │-1│ │2│
W = span(│0│, │1│) ╭╮span(│ 1│, │1│).
│0│ │1│ │ 0│ │2│
└ ┘ └ ┘ └ ┘ └ ┘
7. (10%) Prove that the vector set{u, v, w} is linear independent if and only
if the vector set{u+v, v+w, w+u} is linear independent.
8. (14%)
┌ ┐ ┌ ┐ ┌ ┐ ┌ ┐ ┌ ┐
│ 1│ │1│ │0│ │ 2│ │1│
(a) Show that span(│ 0│, │1│, │1│)=span(│ 1│, │2│)
│-1│ │0│ │1│ │-1│ │1│
└ ┘ └ ┘ └ ┘ └ ┘ └ ┘
(b) Give the linear equation describing the subspace in (a).
9. (16%) Find a basis for each of the four subspaces R(A) = C(A^T), C(A), N(A),
N(A^T) of the matrix (Don't use Row Exchange operation in Gaussian
Elimination to obtain the Reduced Echelon Form).
┌ ┐
│1 3 0 -1 2│
A = │0 1 -2 1 0│
│2 5 3 -4 0│
│3 11 -4 -1 6│
└ ┘
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