課程名稱︰線性代數
課程性質︰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）︰2014/11/30
考試時限（分鐘）：180
試題 :
總共十一題，每題十分，可按任何順序答題。可參考個人事先準備的A4單頁單面的大抄。
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。
第一題
Let T : V → W be linear, where V and W are finite-dimensional vector spaces
over F. If W' is a subspace of W, then the subset V' of V consisting of the
vectors x ∈ V with T(x) ∈ W' is a subspace of V.
第二題
Let V be a vector space with n = dim(V) < ∞. If T : V → V is linear, then
there exist ordered bases α and β of V such that [T]_α^β is a diagonal
matrix in M_nxn(F). (C ∈ M_nxn(F) is diagonal if C_ij = 0_F holds for all
indices i and j with 1 ≦ i ≠ j ≦ n.)
第三題
Let V and W be vector spaces over F. For any subspace U of V, define Z(U)
to be composed of the linear transformations T ∈ L(V,W) such that T(x) = 0_w
holds for each x ∈ U. You are given the fact that Z(U) is a subspace of
L(V,W). If U_1 and U_2 are subspace of V, then
Z(U_1 + U_2) = Z(U_1) ∩ Z(U_2)
第四題
If A, B ∈ M_nxn(F), then
trace(A x B) = trace(B x A).
(Reminder:trace(C) = C_11 + ... + C_nn for any C ∈ M_nxn(F).)
第五題
If T : V → V is linear, where V is a vector space with dim(V) < ∞, then
TT = T_0 if and only if T(V) ⊆ N(T).
(Reminder: TT is the composition of T and T. T_0 is the zero transformation
over V, which maps each vector of V to 0_V. T(V) is the reage space of T.
N(T) is the null space of T.)
第六題
Let V and W be finite-dimensional vector spaces over F. If T_0 不屬於{T_1,T_2}
⊆ L(V,W) with T_1(V) ∩ T_2(V) = {0_w},then {T_1,T_2} is linerly independent.
第七題
Let T : V → W be linear, where V and W are vector spaces over F with dim(V) =
dim(W) < ∞. If β is a basis of V, then
T is an isomorphism if and only if
T(β) is a basis of W.
第八題
If B ∈ M_nxn(F) is invertible, then the transformation
T : M_nxn(F) → M_nxn(F)
defined by
T(A) = B^(-1) x A x B
is an isomorphism.
第九題
Let T : V → W be linear, where V and W are finite-dimensional vector spaces
over F. If (1) α and β are ordered bases of V and (2) γ and δ are ordered
bases of W, then
[T]_β^δ = ([I_W]_δ^γ)^(-1) x [T]_α^γ x [I_V]_β^α
第十題
Suppose A, B ∈ M_mxn(F). If P ∈ M_mxm(F) and Q ∈ M_nxn(F) are invertible
matrices with
B = P^(-1) x A x Q
then there exist (1) vector spaces V and W over F with dim(V) = n and
dim(W) = m,(2) ordered bases α and β of V, (3) ordered bases γ and δ of W,
and (4) a linear T : V → W such that
A = [T]_α^γ and B = [T]_β^δ
第十一題
If A and B are 2x2 matrices over R such that A x B is the zero matrix, then
at least one of A and B is the zero matrix.