Show the independence of three vectors

Question

I took the following question from Zhang, Fuzhen. Linear Algebra (Johns Hopkins Studies in the Mathematical Sciences) . Johns Hopkins University Press.

Show that $\alpha_1 = (1,1,0)$, $\alpha_2 = (1,0,1)$, and $\alpha_3 = (0,1,1)$ form a basis for $\mathbb{R}^3$.

The answer key gives the following solution:

Let $x_1\alpha_1 + x_2\alpha_2 + x_3\alpha_3 = 0$. Then $x_1 + x_2 = 0$, $x_1 + x_3 = 0$, and $x_2 + x_3 = 0$.

Thus $x_1 = x_2 = x_3 = 0$.

Why is $x_1+x_2=0$? Where did these come from?

Thanks,

Answer 1

Because from $x_1\alpha_1+x_2\alpha_2+x_3\alpha_3=0$ we obtain: $$(x_1,x_1,0)+(x_2,0,x_2)+(0,x_3,x_3)=(0,0,0).$$ Can you end it now?