1920t1math1030cw8sol.pdf - 2019-2020 MATH1030 Classwork 8 Last updated November 3 2019 1 Important for students with tutorial on Thursday Nov 7(next Thu

1920t1math1030cw8sol.pdf - 2019-2020 MATH1030 Classwork 8...

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2019-2020 MATH1030 Classwork 8 Last updated: November 3, 2019 1. Important for students with tutorial on Thursday : Nov 7 (next Thu) is the Congregation for Conferment for Bachelor and Master degrees. There is no class on that day. I will post CW8 on Nov 8 Friday 8pm. You can hand in your solution on the Nov 14, during tutorial. 2. Midterm solution has been posted on the main class webpage. For password protected area (lecture notes or exercises): login : math1030 password : like1030 3. online exercise : new question sets Link: click on the course math1030 Login: your student ID Password: your student ID (please change it right after you login) 0.2 pt for each question set . 4. Hand in : Q1, Q2, Q3, Q5, Q6, Q8 5. For section A , Hand in Q1, Q2, Q3, Q5, Q6, Q8 If you have extra time, do Q7, Q9, Q10, Q11, Q12, Q13. You are allowed to skip any computational questions that you think are too easy. But for each skipped question, you have to replace it by one difficult question (Q7, Q10-Q13). Questions 1. Let S = { v 1 = 1 2 - 1 , v 2 = 0 1 0 , v 3 = 0 2 0 , v 4 = 1 3 - 1 , v 5 = 1 4 - 1 , v 6 = 0 1 1 } . Find a subset T of S such that (i) h T i = h S i and (ii) T is linearly independent. Answer. Consider the following matrix which is obtained by juxtaposing the vectors v i in S . 1 0 0 1 1 0 2 1 2 3 4 1 - 1 0 0 - 1 - 1 1 - 2 R 1 + R 2 -----→ R 1 + R 3 1 0 0 1 1 0 0 1 2 1 2 1 0 0 0 0 0 1 - R 3 + R 2 -----→ 1 0 0 1 1 0 0 1 2 1 2 0 0 0 0 0 0 1 1
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Now take T = { v 1 , v 2 , v 6 } . Then by Lecture 13 Theorem 2, h T i = h S i and T is linearly independent. 2. Let A and B be two row equivalent matrices. Let A i (resp. B i ) be the i -th column of A (resp. B ). Recall the following two facts from Lecture 13 Sect 2 (you don’t have to prove the facts). For distinct numbers 1 i 1 , . . . , i k , j n , α i R for i = 1 , . . . , n . Fact 1 : α 1 A i 1 + α 2 A i 2 + · · · + α k A i k = 0 if and only if α 1 B i 1 + α 2 B i 2 + · · · + α k B i k = 0 Fact 2 : α 1 A i 1 + α 2 A i 2 + · · · + α k A i k = A j if and only if α 1 B i 1 + α 2 B i 2 + · · · + α k B i k = B j Suppose a 3 × 5 matrix A row reduces to B = 1 0 2 0 7 0 1 1 0 1 0 0 0 1 - 2 . (a) By the facts, explain why { A 1 , A 2 , A 4 } is linearly independent. (b) By the facts, show that { A 1 , A 2 , A 3 } is linearly dependent. Find a non-trivial relation of linear dependence. (c) By the facts, show that { A 1 , A 2 , A 4 , A 5 } is linearly dependent. Find a non- trivial relation of linear dependence. (d) Let S = { A 1 , . . . , A 5 } . Find a subset T of S such that (i) h T i = h S i and (ii) T is linearly independent. (e) Express A 1 - A 2 + 2 A 3 - 2 A 4 + 3 A 5 as a linear combination of T . (f) Let R = { A 1 , A 3 , A 4 } . i. Show that A 2 is a linear combination of R and hence show that if w is in h T i , then it is in h R i . ii. Show that h R i = h T i = h S i iii. Show that R is linearly independent.
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