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︰2011年11月05日
考試時限：180分鐘
是否需發放獎勵金：是
試題 :
There are problems A to F with a total of 140 points. Please write
down your computational or prrof steps clearly on the answer sheets.
A. Consider the matrix
┌ 0 -1 1 3 3 ┐
│-1 1 1 2 3 │
A = │ 0 2 -2 -6 -6 │
└ 0 0 1 1 2 ┘
(a)(10 points) Find its LDU-decomposition in the form PA=LDU, where
P is a permutation matrix. Find the rank and nullity of A.
(b)(8 points) From PA=LDU, write down a basis for the four fundamental
subspaces of A.
4
(c)(7 points) Given b∈R , find conditions on b so that Ax = b is
solvable, and find the general solutions.
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B.(15 points) Consider the square matrix (where a, b, c, d are real)
┌ a a a a ┐
│ a b b b │
A = │ a b c c │
└ a b c d ┘
-1
Determine the conditions on a,b,c,d s.t. A is invertible, and find A .
When A is not invertible, find its rank and nullity. Do you need
permutation matrix to perform LDU-decomposition of A?
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x
C. Consider the real vector space P = { p(x) e | p(x) is a polynomial
3
of degree <= 3}, and T:P →P defined by T(f) = f' - f.
3 3
(a)(6 points) Find ker(T), and range(T), and verify the nullity theorem.
(b)(14 points)
Find the matrix A which represents T with respect to the basis
3 x 3 x 3 x x
f1(x) = (x-1) e , f2(x) = (x+1) e , f3(x) = x e , f4(x) = e
and the matrix B which also represents T with respect to the basis
3 x 2 x x
g1(x) = (x +x+1)e , g2(x) = (x +x+1)e , g3(x) = (x+1)e ,
x
g4(x) = e . Are A and B similar matrices? Prove your answer.
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D. Given the matrix
┌ -1 1 0 0 ┐
│ -1 0 1 0 │
A = │ 0 -1 1 0 │
│ 0 -1 0 1 │
│ -1 0 0 1 │
└ 0 0 -1 1 ┘
(a)(10 points) Draw the simple connected graph G whose incidence matrix
6
is A. Apply KCL to write independent conditions on b∈R s.t. b lie
in the column space of A. How many independent loops are there in G?
(b)(10 points) The path T (in the underlying graph of G) given by
vertex 1 → vertex 4 → vertex 3 → vertex 2 is a spanning tree of G.
Find a LDU-decomposition of A in the form PA=LDU so that the first 3
rows of U is the incidence matrix of T.
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E. Prove the following statements. Each has 15 points.
(a) Given a block matrix of the form Q = ┌ Ik B ┐ Prove that Q is
└ C Il ┘
invertible iff Ik - BC is invertible iff Il - CB is invertible.
-1
Express Q in terms of B, C explicitly.
(Ik, Il就是k*k, l*l的單位矩陣)
(b) Let C be a binary linear code, and define
ε = { c∈C | c has even weight}. Prove that ε is also a linear
code, and when ε≠ C, the number of elements of ε is exactly half
of the number of elements in C.
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F. Determine which of the following statement is true. Prove your answer.
Each has 6 points.
(a) Let V be a vector space over R. S, T: V → V are two linear
transformations. If T(S(v)) = 100v for all v∈V, then T is onto,
and hence T is invertible.
(b) Any [10,7,d]-binary linear code is not 1-error correcting, but is
possible to be 1-error detecting.
T T
(c) A matrix A has full rank iff either AA is invertible, or A A is.
(d) If a matrix as a LU-decomposition, then its LU-decomposition must
be unique.
8
(e) There exist three 6-dimensional subspaces U, V, W of F such that
dim(U∩V∩W) = 1.
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