I'm given the following question:
Given $v_1, v_2, v_3$ in $\mathbb{R}^5$; $v_2$ not a multiple of $v_1$; $v_3$ not a linear combination of $v_1$ and $v_2$. Must $\{v_1, v_2, v_3\}$ be linearly independent?
There is an equivalent question in the Ch. 1 Supplementary Exercises for Linear Algebra and Its Applications (Lay), the answer being true (the set must be linearly independent).
However, I think that there are cases where such a set could be linearly dependent. For example, the set $${ v_1= (0,0,0,0,0), \hspace{5pt}v_2= (1,1,1,1,2), \hspace{5pt}v_3=(1,1,1,1,3) }$$ The given conditions seem to hold, and $1\cdot v_1 + 0\cdot v_2 + 0\cdot v_3 = (0,0,0,0,0)$.
What do you think?
Linear independence
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linear-algebra
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0I edited you title since it was too long and hard to read. Your'e welcome to revert my changes and name it differently. – 2012-10-17
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1You are right, this example shows that they are not l.i. – 2012-10-17