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10月15日
考試時限：110分鐘
是否需發放獎勵金：是
試題 :
Write your answer on the answer sheet. You should include in your answer every
piece of computation and every piece of reasoning so that corresponding credit
could be gained. You can apply all theorems in Chapter 1, without proof. You
can apply other theorems, by giving a thorough proof.
I. Let W ⊂ R^4 be the subspace consisting of vectors (x1,x2,x3,x4) satisfying
the linear equations x1-x2+x3-x4=0 and 2(x1)+(x2)+2(x3)-(x4)=0. Find a
basis of W. (25 points)
II. Let S={v1,v2,v3,v4,v5} ⊂ R^4, where v1=(1,9,6,4), v2=(9,1,4,6),
v3=(6,4,1,9), v4=(4,6,9,1), v5=(1,-1,0,0). Find a maximal linearly
independent subset of S. (25 points)
III. Give brief answers to the following (5 of each):
(a) Why the empty set is not a subspace of a vector space?
(b) Give an example in which a subset of a vector space V is itself a
vector space but not a subspace of V.
(c) Give an example in which the union of two subspaces of a vector space
is not a subspace.
(d) Give an example in which a subset {v1,v2} in a vector space is linear
independent, while the subset {v1+v2, v1-v2} is linearly dependent.
(e) Why a vector space containing an infinite linearly independent subset
cannot be a finite dimensional vector space?
IV. Give vigorous proofs for the following statements.
(a) (15 points) If W1 and W2 are finite dimensional subspaces of a vector
space, then W1∩W2 and W1+W2 are also finite dimensional, and we have
dim(W1+W2) + dim(W1∩W2) = dimW1 + dimW2
(b) (10 points) Consider the vector space (over R)
(a0) + (a1)x + (a2)x^2 + (a3)x^3
V = {───────────────── | a0,a1,a2,a3 in R}
x^2(x^2 +1)
with the obvious addition and the scalar multiplication. Then
S = { 1/x, 1/x^2, 1/(x^2 + 1) } ⊂ V is linearly independent and
T = { 1/x, 1/x^2, 1/(x^2 + 1), x/(x^2 + 1)} ⊂ V is a basis.
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