Question

Is it possible that vectors v1, v2, v3 are linearly dependent, but the vectors w1 = v1 + v2, w2 = v2 + v3 and w3 = v3 + v1 are linearly independent?

Answer 1

Since $v_1,\ v_2,\ v_3$ are linearly dependent, there exist coefficients $\alpha_1,\ \alpha_2,\ \alpha_3$ such that

$\alpha_1v_1+\alpha_2v_2+\alpha_3v_3=0$

Now, a linear combination of the new vectors would look like this:

$\beta_1w_1+\beta_2w_2+\beta_3w_3 = \beta_1(v_1+v_2)+\beta_2(v_2+v_3)+\beta_3(v_1+v_3)$

Which simplifies to

$(\beta_1+\beta_3)v_1+(\beta_1+\beta_2)v_2+(\beta_2+\beta_3)v_3$

So, any linear combination of $w_1,\ w_2,\ w_3$ is also a linear combination of $v_1,\ v_2,\ v_3$. This implies that we can choose the coefficients for a linear combination that will give the zero vector.

In particular, if $\alpha_1,\ \alpha_2,\ \alpha_3$ are the coefficients such that

$\alpha_1v_1+\alpha_2v_2+\alpha_3v_3=0$

we can choose

$\beta_1+\beta_3=\alpha_1,\quad \beta_1+\beta_2=\alpha_2,\quad \beta_2+\beta_3=\alpha_3$

And we have

$\beta_1w_1+\beta_2w_2+\beta_3w_3 = 0$