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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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.  • Summer '14
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