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August 24th, 2015, 08:42 AM  #1 
Member Joined: Mar 2015 From: Brasil Posts: 90 Thanks: 4  Linear Independence and Linear Dependence.
If $\left \{ u,v \right \}$, $\left \{ v,w \right \}$ and $\left \{ w,u \right \}$ LI are subsets of a vector space V, then $\left \{ u,v,w \right \}$ is LI? Note: LI is Linear Independence. 
August 24th, 2015, 10:19 AM  #2 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,614 Thanks: 2603 Math Focus: Mainly analysis and algebra 
What happens if $\vec u$, $\vec v$ and $\vec w$ are coplanar? Can they be distinct and coplanar?
Last edited by skipjack; August 24th, 2015 at 11:24 AM. 
August 24th, 2015, 11:18 AM  #3 
Math Team Joined: Jan 2015 From: Alabama Posts: 3,264 Thanks: 902 
(1, 0) and (0, 1) are linearly independent. (1, 0) and (1, 1) are linearly independent. (0, 1) and (1, 1) are linearly independent. 
August 24th, 2015, 01:44 PM  #4 
Member Joined: Mar 2015 From: Brasil Posts: 90 Thanks: 4 
No, if it is coplanar they would LD (linear dependent). and q I do now ????

August 24th, 2015, 02:05 PM  #5 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,614 Thanks: 2603 Math Focus: Mainly analysis and algebra 
So you have that, if the three vectors are coplanar, they would be LD. Can you find three coplanar vectors that, pairwise are LI? Hint: see Country Boy's post. Last edited by v8archie; August 24th, 2015 at 02:08 PM. 
August 24th, 2015, 02:10 PM  #6 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,614 Thanks: 2603 Math Focus: Mainly analysis and algebra 
The general case of Country Boy's example would be $\vec w = a\vec u + b \vec v$ where $a$ and $b$ are nonzero real numbers. The main advantage of writing it like this is that we get any number of dimensions for free with the notation.


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