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︰2005/11/5 14:00~17:00
試題 :（註：「屬於」符號因BBS無法顯示，故改以中文寫出。）
Math 201 14410-01 (Linear Algebra) Midterm Exam (11/5/2005)
There are Problems A to D with a total of 110 points. Please write down
your computational, or proof steps clearly on the answer sheets.
A. Work out the following problems.
(a)(20 points) Give the matrix
[ 1 2 0 2 1 ]
A = [-1 -2 1 1 0 ]
[ 1 2 -3 -7 -2 ]
Find a basis for the four fundamental subspaces of A, and determine
their dimensions. Reduce A into a form PA=LU where P is a 3*3 permutation
matrix, L is a 3*3 lower triangular matrix with unit diagonal, and U is a
3*5 row-echelon matrix. Write down explicitly all the pivots you use in
the computation.
(b)(20 points) Consider the square matrix
[ 1 v1 0 0 ]
A = [ 0 v2 0 0 ]
[ 0 v3 1 0 ]
[ 0 v4 0 1 ]
(1) Factor A into LDU decomposition, and determine the condition that this
matrix is invertible. -1
(2) When it is invertible, use Gauss-Jordan method to find its inverse A .
When it is not invertible, do you need permutation matrix to reduce A
into row-echelon form?
(3) If A is not invertible, find its rand and nullity. Determine the
4
condition on the vector b 屬於 R such that Ax=b has at least one
solution.
B. Prove the following statements.
(a)(7 points) Let A,B be square matrics. If I-BA is invertible, prove that
-1
I-AB is invertible. Find (I-AB) explicitly.
(b)(10 points) Let V be a vector space over a scalar field. If U,W 屬於 V are
two finite-dimensional, and
dim (U) + dim (W) = dim (U∩W) + dim (U+W)
2
(c)(7 points) Let A be a square matrix. Prove that the nullspace of A is
2
contains the nullspace of A, while the column space of A is contained in
the column space of A.
C. Determine which of the following statements is true or false. If true,
prove it. If false, find a counterexample. Each has 6 points.
(a) If the column vectors of a matrix are independent, the its row vectors are
also independent.
(b) There exists a 3*2 matrix A such that two 2*3 matrices B and C exist, and
satisfy AB=I_3 and CA=I_2
2 2 2
(c) For two square matrices A,B, (AB) =A B iff AB=BA
(d) Let V be a vector space. f：V→V and g：V→V are two linear
transformations. If g(f(v))=2v for all v 屬於 V, then f is one-to-one and
onto.
4
(e) Every subspace of R is the nullspace of some matrix.
(f) The matrix A has the LU decomposition
[ 1 0 0 0 ][ 1 2 0 1 2 1 ]
A = LU = [ 2 1 0 0 ][ 0 0 2 2 0 0 ]
[ 2 1 1 0 ][ 0 0 0 0 0 1 ]
[ 3 2 4 1 ][ 0 0 0 0 0 0 ]
Then columns 1,3,6 of U are a basis for the column space of A.
m
D.(10 points) If a symmetric square matrix A satisfy A =0 for some positive
integer m, prove that A=0.
--
※ 發信站: 批踢踢實業坊(ptt.cc)
◆ From: 140.112.240.198