[試題] 106-1 呂學一 線性代數 第二次期中考消失

看板NTU-Exam作者時間6年前 (2017/11/28 22:16), 6年前編輯推噓2(2推 0噓 0→)
留言2則, 2人參與, 最新討論串1/1
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）︰2017/11/28 考試時限（分鐘）：180 試題 : 共十二題每題十分，可按任意順序答題。每題難度不同，請判斷恰當的解題順序。 可直接使用課堂上證明過或出現過的任何定理或性質。 使用時請標示定理或性質的名稱或編號。 ---------------------------------------------------------------------------- 第一題 Suppose T∈L(V) for a vector space V with dim(V) < ∞. Prove or disprove that 2 if T is invertible, then so is T. ---------------------------------------------------------------------------- 第二題 Let T∈L(V) for a vector space V. Prove or disprove that 2 T = T if and only if T(V)⊆N(T). 0 ---------------------------------------------------------------------------- 第三題 Let V be a vector space over scalar field F with dim(B) < ∞. Let T_1 and T_2 be distinct members of L(V). Prove or disprove that if T_1(V) ∩ T_2(V) = {0_V} then {T_1 , T_2} is a linearly independent subset of the vector space over F. ---------------------------------------------------------------------------- 第四題 Let T∈L(V) for a finite-dimensional vecter space V. Prove or disprove that N(T) + T(V) = V if and only if N(T) ∩ T(V) = {0_V} ---------------------------------------------------------------------------- 第五題 Let {x_1 , x_2 , ... , x_n} be a basis of a vector space V with dim(V) = n. Prove or disprove that {y_1 , y_2 , ... , y_n} is a basis of V , where 　　def y_i　＝ x_1 + x_2 + ... + x_i for each i∈{1,2,...,n}. ---------------------------------------------------------------------------- 第六題 Let T ∈ |F(V) for a finite-dimensional vector space V over Q (有理數). Prove or disprove that if T(x+y) = T(x) + T(y) holds for any vectors x and y of V , then T ∈ L(V). ---------------------------------------------------------------------------- 第七題 Let T ∈ |F(V) for a finite-dimensional vector space V over C (複數). Prove or disprove that if T(x+y) = T(x) + T(y) holds for any vectors x and y of V , then T ∈ L(V). ---------------------------------------------------------------------------- 第八題 For any given q∈Q and n ≧ 0, let f(q,n) be the dimemsion of the subspace {p(x) ∈ |P_n(|R) | p(q) = 0} of the vector space |P_n(|R) over |R. Give the formula for f(q,n). Justify your answer. 　 ---------------------------------------------------------------------------- 第九題 Suppose T∈L(V) for a vector space V with dim(V) < ∞. Prove or disprove that if U is a subspace of V with T(U) ⊆ U, then U = N(T) ---------------------------------------------------------------------------- 第十題 Suppose T∈L(V) for a vector space V with dim(V) < ∞. Prove or disprove that there always exists a positive integer k with k k V = N(T ) ⊕ T (V) ---------------------------------------------------------------------------- 第十一題 Let U_1 , U_2 , V_1 , ... , V_j be subspaces of a finite dimensional vector space W satisfying U_1 = V_1 ⊕ ... ⊕ V_i U_2 = V_(i+1) ⊕ ... ⊕ V_j for some 1≦i＜j. Prove or disprove that if U_1∩U_2 = {0_W} then U_1 + U_2 = V_1 ⊕ ... ⊕V_j ---------------------------------------------------------------------------- 第十二題 Suppose that T_1 and T_2 are members of L(V) for a finite-dimensional vector space V. Prove or disprove that if T_1T_2 = I_V then T_2T_1 = I_V ---------------------------------------------------------------------------- -- ※ 發信站: 批踢踢實業坊(ptt.cc), 來自: 140.112.211.204 ※ 文章網址: https://www.ptt.cc/bbs/NTU-Exam/M.1511878573.A.735.html ※ 編輯: bluedog666 (140.112.211.204), 11/28/2017 22:21:10
推
11/29 00:17, , 1F
已收資訊系精華區!
11/29 00:17, 1F
※ 編輯: bluedog666 (140.112.211.204), 11/29/2017 21:58:46
推
12/02 22:56, , 2F
好奇 T_0是指0映射嗎？
12/02 22:56, 2F
對的~ ※ 編輯: bluedog666 (140.112.211.204), 12/18/2017 23:57:04
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