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August 24th, 2015, 09:28 AM   #1
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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.
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August 24th, 2015, 11:45 AM   #2
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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 non-zero 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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