From Linear Algebra Done Wrong. Don't know how to proof or disprove this.

Is it possible that vectors $\mathbf{v_1}$ ,$\mathbf{v_2}$ ,$\mathbf{v_3}$ are linearly dependent, but the vectors
$\mathbf{w_1} = \mathbf{v_1} + \mathbf{v_2}$ , $\mathbf{w_2} = \mathbf{v_2} + \mathbf{v_3}$ and $\mathbf{w_1} = \mathbf{v_3} + \mathbf{v_1}$ are linearly independent?

Question

From Linear Algebra Done Wrong.  Don't know how to proof or disprove this.
 
Is it possible that vectors $\mathbf{v_1}$ ,$\mathbf{v_2}$ ,$\mathbf{v_3}$ are linearly dependent, but the vectors
$\mathbf{w_1} = \mathbf{v_1} + \mathbf{v_2}$ , $\mathbf{w_2} = \mathbf{v_2} + \mathbf{v_3}$ and $\mathbf{w_1} = \mathbf{v_3} + \mathbf{v_1}$ are linearly independent?

mathmate · Answer

If $v_1,\ v_2, \ v_3$ are dependent, then we can write
$v_3=p\ v_1+q\ v_2$ by definition of dependence, where p, q are some constants to be determined.

Then the system of equations $w_1=..., w_2=..., w_3=...$
can be expressed in only two independent variables, $v_1$ and $v_2$, which means that  $w_1$,$w_2$, and $w_3$ must be linear dependent as well.