Since [tex]v_1,\ v_2,\ v_3[/tex] are linearly dependent, there exist coefficients [tex]\alpha_1,\ \alpha_2,\ \alpha_3[/tex] such that
[tex]\alpha_1v_1+\alpha_2v_2+\alpha_3v_3=0[/tex]
Now, a linear combination of the new vectors would look like this:
[tex]\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)[/tex]
Which simplifies to
[tex](\beta_1+\beta_3)v_1+(\beta_1+\beta_2)v_2+(\beta_2+\beta_3)v_3[/tex]
So, any linear combination of [tex]w_1,\ w_2,\ w_3[/tex] is also a linear combination of [tex]v_1,\ v_2,\ v_3[/tex]. This implies that we can choose the coefficients for a linear combination that will give the zero vector.
In particular, if [tex]\alpha_1,\ \alpha_2,\ \alpha_3[/tex] are the coefficients such that
[tex]\alpha_1v_1+\alpha_2v_2+\alpha_3v_3=0[/tex]
we can choose
[tex]\beta_1+\beta_3=\alpha_1,\quad \beta_1+\beta_2=\alpha_2,\quad \beta_2+\beta_3=\alpha_3[/tex]
And we have
[tex]\beta_1w_1+\beta_2w_2+\beta_3w_3 = 0[/tex]
