Proving a set of vectors is independent

Question

Firstly, my apologies for attaching the question as an image. I was having a lot of problems typing it in MathJax.
The question asks to show S is independent. From my understanding, I have to prove that:
$$ \alpha{\begin{pmatrix} 1\\ 1\\ 0\\ \end{pmatrix}} +\beta{\begin{pmatrix} 1\\ 0\\ 1\\ \end{pmatrix}}= {\begin{pmatrix} 0\\ 0\\ 0\\ \end{pmatrix}} $$if and only if $\alpha$ =$\beta$ = 0. I try to solve this using an augmented matrix, and reduce it to row-echelon form. $$ \left[ \begin{array}{cc|c} 1&1&0\\ 1&0&0\\ 0&1&0 \end{array} \right] $$ $\underrightarrow{R2+R3}$ $$ \left[ \begin{array}{cc|c} 1&1&0\\ 1&1&0\\ 0&1&0 \end{array} \right] $$ $\underrightarrow{R3-R2}$ $$ \left[ \begin{array}{cc|c} 1&1&0\\ 1&1&0\\ 0&0&0 \end{array} \right] $$ But, $$ \left[ \begin{array}{cc|c} 0&0&0\\ \end{array} \right] $$ means that the system has infinitely many solutions. Therefore, $\alpha,\beta$ could have infinitely many solutions, and so the set $S$ is dependent.
But the solution says that the set is independent because $\alpha,\beta = 0$. Could someone please assist me by pointing out where I have gone wrong? I would appreciate if you could stick to my method of solving (proving $\alpha,\beta=0$) because this is how I would like to approach it)

Answer 1

From $$ \alpha{\begin{pmatrix} 1\\ 1\\ 0\\ \end{pmatrix}} +\beta{\begin{pmatrix} 1\\ 0\\ 1\\ \end{pmatrix}}= {\begin{pmatrix} 0\\ 0\\ 0\\ \end{pmatrix}}$$

If you consider the third coordinate, you directly get $\beta=0$.

Answer 2

$$\alpha \left(\begin{matrix}1\\1\\0\end{matrix}\right)+\beta \left(\begin{matrix}1\\0\\1\end{matrix}\right)= \left(\begin{matrix}0\\0\\0\end{matrix}\right)\\\implies\begin{cases}\alpha \times 1+\beta\times 1=0\\\\\alpha \times 1+\beta\times 0=0\\\\\alpha \times 0+\beta\times 1=0\end{cases}\\\implies \alpha =\beta=0$$