課程名稱︰線性代數一
課程性質︰數學系必修
課程教師︰陳其誠
開課學院：理學院
開課系所︰數學系
考試日期︰2010年12月24日
考試時限：100分鐘
是否需發放獎勵金：是
試題 :
Write your answer on the answer sheet.
1.(25 points) Solve the following system of linear equations.
2x+2y+v=0
x+y-w=0
z+3w=0
x+y-z-4w=0
x+y+z+v+4w=0
2.(25 points) Calculate det(A), where
┌ ┐
│1 2 3 4 5│
│2 3 4 5 1│
│3 4 5 1 2│
│4 5 1 2 3│
│5 1 2 3 4│
└ ┘
3. Either give a brief reason or give a counter example for each of the
following assertions (5 points each):
(a) If (a1,a2,a3,a4), (b1,b2,b3,b4)∈R^4 are linearly independent, then
4 4
the solution set W of the system of equations Σaixi=0 and Σbixi=0
i=1 i=1
is a 2-dimensional subspace of R^4.
(b) If W is a 2-dimensional subspace of R^4, then there exist linearly
independent (a1,a2,a3,a4), (b1,b2,b3,b4)∈R^4 such that W is the
4 4
solution set of the system of equations Σaixi=0 and Σbixi=0
i=1 i=1
(c) An n-linear function f:M (F)→F must be linear.
n*n
(d) Suppose f,g:M (F)→F are alternating n-linear functions with f(I)=g(I),
n*n
then f(A)=g(A) for all A.
(e) If two n*n matrices A and B have the same reduced row echelon form,
then they must be similar.
4. Give a rigorous proof for (a) and answer (b) with rigorous reasoning.
(a)(15 points) For any A∈M (F), det(A^t)=det(A).
n*n
(b)(10 points) Let e1,...,e5 be the standard basis of R^5. It is known that
fi=ei+e4, i=1,2,3,5, form a basis of
W = {(x,y,z,v,w)∈R^5 | x+y+z-v+w=0 }
Denote a=(1,2,3,10,4)∈W. Find all subsets {b,c,d}⊂{f1,f2,f3,f5} such
that the vectors a,b,c,d form a basis of W (give the complete list of
all these subsets).
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