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August 24th, 2015, 09:28 AM  #1 
Member Joined: Mar 2015 From: Brasil Posts: 90 Thanks: 4  Linear Independence and Linear Dependence.
Prove that the three vectors are $\mathbb{R}^{2}$ LD (Linear Dependence). Generalize this result to $\mathbb{R}^{n}$.
Last edited by skipjack; August 24th, 2015 at 12:25 PM. 
August 24th, 2015, 11:45 AM  #2 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,146 Thanks: 2386 Math Focus: Mainly analysis and algebra 
Suppose we are given vectors $\vec u$, $\vec v$ and $\vec w$ all in $\mathbb R^2$. If $\vec u$ and $\vec v$ are LD then there exist nonzero real numbers $a$ and $b$ such that $a\vec u + b\vec v = 0$. And we can easily expand this linear combination to include $\vec w$ too. If $\vec u$ and $\vec v$ are LI, they must span $\mathbb R^2$ (do you need to prove this?) and so $\vec w = a\vec u + b\vec v$ for some real numbers $a$ and $b$. 

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dependence, independence, linear 
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