課程名稱︰線性代數
課程性質︰資訊系必修
課程教師︰呂學一
開課學院：電機資訊學院
開課系所︰資訊工程學系
考試日期（年月日）︰2014/10/26
考試時限（分鐘）：180
是否需發放獎勵金：是
（如未明確表示，則不予發放）
試題 :
總共十二題，每題十分，可按任何順序答題。只能參考個人事先準備的A4單頁單面大抄。
每一題都是一個可能對也可能不對的敘述。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(第十二
題除外)。
第一題
If V is a subset of W and and (W,F,+,‧) is vector space, then V is a subspace
of (W,F,+,‧) if and only if V = span(V).
第二題
Recall that W = (F(F_1,F_2),F_2,+,‧) is a vector space, where F_1 and F_2 are
fields. If (1) U consists of the functions f ∈ F(F_1,F_2) with f(-a) = f(a)
for all a ∈ F_1 and (2) V consists of the functions g ∈ F(F_1,F_2) with
g(-a) = -g(a) for all a ∈ F_1, then U∪V is a subspace of W.
第三題
If (W,F,+,‧) is a vector space and V is a subset of W, then V is a subspace
of (W,F,+,‧) if and only if -x + ay ∈ V holds for any scalar a ∈ F and any
vectors x,y ∈ V.
第四題
Let V_1 and V_2 be subspaces of vector space (W,F,+,‧). Let V = V_1 + V_2. If
V_1∩V_2 = {0_W}, then for each vector x ∈ V there is a unique pair (x_1,x_2)
with x_1 ∈ V_1, x_2 ∈ V_2, and x = x_1 + x_2.
第五題
If (U,F,+_U,‧_U) and (V,F,+_V,‧_V) are two vector spaces over scalar field F,
then so is the algebraic structure (W,F,+_W,‧_W) defined as follows, where
a ∈ F, u_1,u_2 ∈ U, and v_1,v_2 ∈ V:
W = {(u,v) : u ∈ U, v ∈ V}
(u_1,v_1) +_W (u_2,v_2) = (u_1 +_U u_2, v_1 +_V v_2)
a ‧_W (u_1,v_1) = (a ‧_U u_1, a ‧_V v_1).
第六題
Let S = {x_1,...,x_n} with 1 ≦ n = |S| < ∞ be a linearly independent subset
of vector space (V,F,+,‧). For each vector x ∈ span(S), there exists a unique
n-tuple (a_1,...,a_n) ∈ F^n such that x = a_1 x_1 + ... + a_n x_n.
第七題
If x and y are two distinct vectors of vector space (V,F,+,‧), then {x,y} is
linearly dependent if and only if there exists a scalar a ∈ F\{1_F} such that
x = ay or y = ax.
第八題
If S with |S| = ∞ is a subset of vector space V = (V,F,+,‧), then S is
linearly independent if and only if each finite subset of S is linearly
independent.
第九題
If S with |S| = ∞ is a spanning subset of vector space V = (V,F,+,‧) with
dim(V) < ∞, then there is a basis B of V with B ⊆ S.
第十題
If U and V are finite-dimensional subspaces of vector space (W,F,+,‧), then
dim(U+V) = dim(U) + dim(V) - dim(U∩V).
第十一題
If T : M_9x9 (R) → R is defined by letting T(A) be the sum of diagonal entries
of matrix A, then T is linear.
第十二題
Let V and W be two vector spaces over a common scalar field. If T : V → W is
linear, then T(0_V) = 0_W.