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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: 7,305 Thanks: 2443 Math Focus: Mainly analysis and algebra What happens if $\vec u$, $\vec v$ and $\vec w$ are co-planar? Can they be distinct and co-planar? 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: 3,159 Thanks: 866 (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 co-planar 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: 7,305 Thanks: 2443 Math Focus: Mainly analysis and algebra So you have that, if the three vectors are co-planar, they would be LD. Can you find three co-planar 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: 7,305 Thanks: 2443 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 non-zero 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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