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Suppose $a,b$ and $c$ are linearly independent vectors in a vector space $V$. How can I prove that $a+b$ or $b+c$ are also linearly independent?

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Linearly independent w.r.t which vectors? Obviously not $a,b$ and $b,c$. – Jacob Sep 30 '12 at 16:31
I think what is intended is that you show that the set $\{a+b,b+c\}$ is a linearly independent set. – André Nicolas Sep 30 '12 at 16:32
Well that's the problem. the question is just giving a,b and c. no actual values. – Nima Sep 30 '12 at 16:33
You don't need actual values-the problem is that $a+b$ and $b+c$ are just single vectors, and linear independence of one vector on its own is trivial. So as @AndréNicolas said a more reasonable question is to prove that $a+b$ and $b+c$ are linearly independent, as was proved below. – Kevin Carlson Sep 30 '12 at 16:36
Thanks a lot guys. – Nima Sep 30 '12 at 16:37
up vote 7 down vote accepted

If I understand your question correctly, you want to show that if $a$, $b$, $c$ are linearly independent, then $a+b$ and $b+c$ are linearly independent.

Just look at the definitions.

You know that $x_1a+x_2b+x_3c=0$ implies $x_1=x_2=x_3=0$. (This is the definition of linear independence for three vectors.)

You ask whether $y_1(a+b)+y_2(b+c)=0$ implies $y_1=y_2=0$.

Just simplify this to get: $y_1 a + (y_1+y_2)b +y_2c=0$. This implies that $y_1=y_1+y_2=y_2=0$. The condition $y_1+y_2=0$ is redundant there, but we have shown that $y_1=y_2=0$.

This means that the vectors $a+b$, $b+c$ are linearly independent.

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