[試題] 102上 線性代數 第一次期中考 呂學一消失
課程名稱︰線性代數
課程性質︰資工系 大二上 必修
課程教師︰呂學一
開課學院：電機資訊學院
開課系所︰資工系
考試日期（年月日）︰2013/11/8
考試時限（分鐘）：180(mins)
是否需發放獎勵金：是
（如未明確表示，則不予發放）
試題 :
總共12題，每題十分，可按任何順序答題。只能參考個人事先準備好的A4單頁單面大抄。
每個小題都是可能對或是不對的敘述。如果你覺得對，請證明它是對的，如果你覺得不對
，請證明它是錯的。課堂上證過的定理，或是提過的練習題，都可以直接引用。
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第一題 If (W,F,+,‧) is a vector space and V⊆W, then V is a subspace of
(W,F,+,‧) if and only if V!=Øand ax+y∈V holds for any scalar a∈F and any
vectors x,y∈V.
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第二題 Let V1 and V2 be subspaces of a vector space (W,F,+,‧). Then, V1∪V2
is a subspace of (W,F,+,‧) if and only if V1⊆V2 or V2⊆V1.
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第三題 If V is a subset of W and (W,F,+,‧) is a vector space, then V is a
subspace of (W,F,+,‧) if and only if V=span(V).
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第四題 If S1⊆S2 are subsets of vector space (V,F,+,‧) and span(S1)=V, then
span(S2)=V.
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第五題 {x^3-x, 2*x^2+4, (-2)*x^3+3*x^2+2*x+6} is linearly independent in
(P_3(R),R,+,‧).
(按) + and ‧ in (P_3(R),R,+,‧) are the operators defined in polynomial.
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第六題 Let S be the subset {(1_F,1_F,0_F),(1_F,0_F,1_F),(0_F,1_F,1_F)} of
vector space (F^3,F,+,‧). (a)If F=R(實數), then S is linearly independent.
(b)If 1_F+1_F=0_F, then S is linearly dependent.
(按) 1_F and 0_F denote the identity elements for multiplication and addition
in scalar field F, respectively.
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第七題 Let f and g be two functions from R to R(i.e., f,g∈F(R,R)) defined by
f(t)=e^(r*t) and g(t)=e(s*t), where r and s are two distinct real numbers.
Then, {f,g} is linearly independent in vector space (F(R,R),R,+,‧).
(按) R in this question indicates 實數.
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第八題 Let S be a infinite spanning set of a finite dimensional vector space
(V,F,+,‧). That is, |S|=∞, dim(V)<∞, and span(S)=V. Then, there is a basis
R of (V,F,+,‧) with R⊆S.
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第九題 Let V1 and V2 be subspaces of vector space (W,F,+,‧) with
dim(V1)<=dim(V2)<∞. (a)dim(V1∩V2)<=dim(V1). (b)dim(V1+V2)<=dim(V1)+dim(V2).
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第十題 (a)There is a linear transformation T:R^2->R^3 with T(1,1)=(1,0,2) and
T(2,3)=(1,-1,4). (b)There are no linear transformations T:R^3->R^2 with
T(1,0,3)=(1,1) and T(-2,0,-6)=(2,1).
(按) R in this question indicates 實數.
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第十一題 Let T:V->W be linear, where V and W are vector spaces over scalar
field F. Let S be a subset of V. (a)Even if S is linearly independent, T(S) may
still be linearly dependent. (b)Even if S is linearly dependent, T(S) may
still be linearly independent.
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第十二題 Let V and W be finite-dimensional vector spaces over scalar field F.
Let T:V->W be linear. (a)If T is onto, then dim(V)>=dim(W). (b)If T is
one-to-one, then dim(V)<=dim(W).
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