103學年度資訊工程系碩士班招生試題

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固 立 臺灣科技 大 學 10 3 學 年 度碩 士班 招生試題 資 訊工 程系碩 士班

計算機數學

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科目 (總分為 1 00 分)

Total Score: 100

1. ( 15%) Let Vj = (4， 6， 7)弋 V2 = (0， 1 ， 1 )弋 V 3 = ( 1 ， 1 ， 2)弋 and let U] = ( 1 ， 1 ， 1 )弋 U 2 = (1 , 2 , 2) T , U 3 = (2 , 3 , 4) ^T

(a) Find 七he transition matrix from { 呵， V 2 , V 3} to {Uj , U 2 , U 3}. (10%)

(b) If x=2vj+ V2 一 V3 ， determine the coord inate vector of x w仙 respect to {U 1 , U 2 , U 3}. (5%)

2 (15%) Let

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(a ) What is t he d削ance between t he vector b and the row spαce of A. (10%)

(b) What is the distance between the vector c and J le column 咖ce ^{of A}

(5%)

3. (20%) Answer each of the stat ements tha七 follow as true or false. Y ou have t o prove 1 t

f your answer is true , ot herwise yo u need to give a co unterexample for it P lease note thιt you will N OT get any point if you only α吼叫ler true or false (a) Let A be an m xηmatrix with m > ^n. Suppose t hat t he ra叫 of A is n

then Ax = b is consiste且七 for a叮 b ε Rm. (5%)

(b) Let Q be a n n x n orthogonal matrix. Then IIQxl12 = II x lk (5%)

(c ) Let A and B be 10叩er triangular matrix and upper triangular matrix w岫 the same diagonal and A is in切rtible. T hen C = AB is invertible. (5%) (d) If λ lS an e耶nvalue of a matrix A , thenλk is a n ei ge盯alue of t he mat rix

A ^k , where k is a pos itive intege r. (5%)

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第 國立臺灣科技大學 103 學年度碩士班招生試題 資訊 工程系碩士班

計算機數學 (總分為 100 分 )

4. (1 5%) (a) Prove 七hat a complete graph Kn , where n 主 3 bei月 a prime number , can have its edges partitioned 叫o ~(n-1) di日:joint Hamil七on cu.cu出 (10%)

(b) Duri月 a summer camp , if a group of 19 students have dimler together

的 a circular table everyday , and if each night each student must si七日ext

to a completely difIe泊的 pair of s七udents ， then how many days can the summer camp last? (5%)

5. (10%) L的口 = 37 x 73. What is the probability that a randomly selected positive integer m , where m <爪 is relatively prime to n?

6. (13%) A robot is trying 七o enter a room through a door that opens at ra吋OlIl. Sup pose the door opens two-thirds of 七he time. Furthermore ，七he robot is able to correctly sense an opened door with probability 0.9 , and correctly s巴nse a closed door with probability 0.7. What is the probability that the door is ac- tually open when the robo七 senses an opened door? ^P

7. (υ12%) P

i凶 nclud巴d in every min 口limum spa 缸 ωI 且1I且1lI且19 tr巴e of G