課程名稱︰線性代數
課程性質︰必修
課程教師︰鄭振牟
開課學院：管理學院
開課系所︰工商管理學系科技管理組
考試日期（年月日）︰2009.10.30
考試時限（分鐘）：100
是否需發放獎勵金：是
（如未明確表示，則不予發放）
試題 :
(以下以€代替屬於，€/代替不屬於)
1. Let V be a vector space, and let S be a nonempty subset of V. The subspace
generated by S is the smallest subspace U containing S. Here U being
smallest means that whenever a subspace contains S, it must contain U, or U
is "between" S and all subspaces containing S. In other words, U is the
intersection of all subspaces containing S, i.e.,
U = ∩ Ui,
S╭ U ≦V
╰ i
where U ≦V means that U is a subspace of V. You have already proved that
i i
U is indeed a subspace of V in Problem 8 of Homework 1.
(a) (10%) Let U and U be two subspaces of V. Explain why U +U is the
1 2 1 2
subspace generated by U and U .
1 2
(b) (10%) Let (u ,...,u ) be a basis of a subspace U of V. Explain why
1 2
{u ,...,u } is a smallest generating set of U, i.e., no sets containing
1 n
strictly less than n vectors can generate U.
2. Let U and U be two subspaces of R^3.
1 2
(a) (7%) Give an example where U ∪U is a subspace of R^3.
1 2
(b) (7%) Give another example where U ∪U is not a subspace of R^3.
1 2
(c) (10%) Suppose U ⊕U = R^3. Give an example where U '≠U but
1 2 1 1
U '⊕U = R^3.
1 2
3. (10%) Let U be a subspace of a vector space V, and let (u ,...,u ) be a
1 n
basis of U. Prove that if w€V\U (this means that w€V but w€/U), then
(u +w,...,u +w) is linearly independent.
1 n
4. Let V and W be two vector spaces over a field F, and let W^V be the set of
all functions (not necessarily linear) from V to W. For any two functions
f and g in W^V, the sum f+g is defined "pointwise" such that (f+g)(v) =
f(v) + g(v) for all v€V. Similarly, for any function f in W^V and any
scalar a in F, the scalar product af is defined such that (af)(v) = af(v)
for all v€V. Obviously, the addition and scalar multiplication so defined
are commutative and associative and furthermore satisfy the distributive
property.
(a) (10%) Prove that together with the addition and scalar multiplication
defined above, W^V is a vector space (not necessarily finite
dimensional).
(b) (10%) Prove that L(V,W) is a subspace of W^V.
(c) (6%) Let V = W = R^3. Show that L(V,W) is a proper subspace of W^V by
giving an example in W^V\L(V,W).
5. (10%) Prove that if (v ,...,v ) spans V and T€L(V,W) is surjective, then
1 n
(Tv ,...,Tv ) spans W.
1 n
6. (10%) Let V and W be two vector spaces, and let T€L(V,W). Prove that if
(Tv ,...,Tv ) is linearly independent in W whenever (v ,...,v ) is linearly
1 n 1 n
independent in V, then T is injective.
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