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Suppose that $\vec{u_1}, \vec{u_2}, \vec{u_3}$ are linearly dependent. What can you say about the linear dependence or independence of the vectors $\vec{v_1} = 2\vec{u_1} + \vec{u_2}$ and $\vec{v_2} = 2\vec{u_2} + \vec{u_3}$ ?

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  • $\begingroup$ Can you use the definition and make an attempt ? $\endgroup$ – Shailesh Mar 21 '16 at 15:44
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If the three vectors are linearly dependent, any linear combination of them will also be dependent on the old vectors ($\vec{v_1}$ and $\vec{v_2}$ are dependent on $\vec{u_1}$, $\vec{u_2}$ and $\vec{u_3}$). However, it does not mean that the new vectors $\vec{v_1}$ and $\vec{v_2}$ will be in general linearly dependent between them.

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Hint

If $u_1, u_2, u_3$ are linearly dependent, then are the linear combinations of them linear dependent?

That is that we know: $$\exists a_1 \neq 0 \lor a_2 \neq 0 \lor a_3 \neq 0, a_1u_1 + a_2u_2 + a_3u+3 = 0$$ So we have some $$a_nu_n = -a_1u_j - a_2u_k$$ where $j, k$ are the other u's that are not $n$.

So what does that mean about if there exists $a_1, a_2, a_3$ for their linear combinations?

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