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 October 5th, 2019, 02:56 AM #1 Newbie   Joined: Oct 2019 From: Hong Kong Posts: 1 Thanks: 0 Span and linear dependence Dear everyone, l really get confused when reading some textbooks on linear algebra : Suppose there are two vectors in R3, u= (3,1,0) and v = (1,6,0). Firstly, u and v are linearly independent because neither vector is a multiple of the other. IF w is a linear combination of u and v, then {u,v,w} is linearly dependent and w is in span {u,v}. So , it joined to conclusion that any set {u,v,w} in R3 with u, and v linearly independent. The set {u,v,w} will be linearly dependent if and only if w is in the plane spanned by u and v . l wonder if this theorem is true if x[SUB]3[/SUB] is not equal to 0 for the above case. (That is not a x1x2 plane with x3 =0) Besides, are the following statements true ? 1) If x and y are linearly independent, and if {x,y,z} is linearly dependent, then z is in Span {x,y} 2) lf x and y are linearly independent , and if z is in Span {x,y}, then {x,y,z} is linearly dependent. Now for a practice problem: There are four vectors u= (3,2,-4) v= (-6,1,7) w= (0,-5,2) z= (3,7, -5) It shows that each pairs of the above vectors are linearly independent because neither vector is a multiple of the other. But, {u,v,w,z} is linearly dependent, because there are more vectors than entries in them. So my question arised, how could{u,v,w,z} be linearly dependent when w is NOT in span{u,v,z} ? According to the definition of linearly dependent, the vector set need to have at least one vector which is a linear combination of the others. So is the above example means something like u is in span {v,z,w} or v = x1u+ x2z + x3w? October 5th, 2019, 06:01 AM #2 Global Moderator   Joined: Dec 2006 Posts: 21,035 Thanks: 2271 In your example, z = 3u + v. Tags dependence, linear, span Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post shashank dwivedi Linear Algebra 5 May 16th, 2018 01:04 PM hyperbola Linear Algebra 1 October 19th, 2015 04:57 AM Luiz Linear Algebra 1 August 26th, 2015 09:22 AM Luiz Linear Algebra 5 August 24th, 2015 02:10 PM Luiz Linear Algebra 1 August 24th, 2015 10:45 AM

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