課程名稱︰線性代數一
課程性質︰數學系必修
課程教師︰林紹雄
開課學院：理學院
開課系所︰數學系
考試日期︰2007年11月11日
考試時限：180分鐘
是否需發放獎勵金：是
試題 :
130 points.
A. Consider the matrix
┌0 0 1 -3 2┐
A = ｜2 -1 4 2 1｜
｜4 -2 9 1 4｜
└2 -1 5 -1 5┘
(a) 15 points Find its LDU-decomposition in the form PA=LDU.Write down
explicitly all the pivots you use in the computation.
(b) 10 points From the form of (a),write down a basis for the four
foundamental subspaces of A.Determine its rank and nullity.
(c) 10 point Consider the system of equations Ax=b for b屬於R^4.Which
variable in x are pivot variables,or free variables?Find the consistency
conditions for b so that Ax=b has at least one solution,and find all its
solutions.
B. 20 points Consider the matrix
┌1 v1 0 0┐
A = ｜0 v2 0 0｜
｜0 v3 1 0｜
└0 v4 0 1┘
where v1 v2 v3 v4 are four given real numbers. Apply the Gauss-Jordan method
to find the condition that A is invertible,and find its inverse explicitly.
When A is not invertible,find the dimentions of its four foundamental
subspaces.
C. Let the subspace V of R^4 be spanned by the following vectors
v1=[1 -1 0 0]^T v2=[1 0 -1 0]^T v3=[1 0 0 -1]^T v4=[0 1 -1 0]^T
v5=[0 1 0 -1]^T v6=[0 0 1 -1]^T
(a) 10 points Find the dimension of V, and extract a basis of V from these
vectors
(b) 15 points Find a linear transformation T:R^4→R^5 so that its range space
T(R^4) is the subspace of R^5 defined by the system of equations
x1+3x2-2x4+x5=0, x2-x3-x4+5x5=0
and T satisfies V∩ker(T)={0} and V∪ker(T)=R^4
Write down the matrix representation of T with respect to the natural basis
of R^4 and R^5.
D.Prove th following ststements.
(a) 8 points
Let A,B be two n*n matrices.If In-BA is invertible,prove that
In-AB is invertible.Find (In-AB)^-1 explicitly.
(b) 10 points
Let V be a finite-dimensional vector space.W⊂V is a subspace. Proce
that dim(V/W)=dim(V)-dim(W)
(c) 8 points
Let A,B be m*n and n*k matrices respectively.If AB=0,prove that
rank(A)+rank(B)≦n.Can the equality be achieved?
E. Determine which of the following statement is true.Prove your answer,or
give a counterexample.Each has 6 points.
(a) The column vectors of square matrix are linearly independent iff its
row vectors are linearly independent.
(b) Let V be a vector space. S,T:V→V are two linear transformations.If
T(S(v))=100v for all v∈V,then T is one-to-one and onto.
(c) 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 the columns1,3,6 of U are a basis for the column space of A
(d) Let A be a m*n real matrix.THen the rank of A achieve the maximum
possible rank iff either (A^T)A or A(A^T) is invertible.
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