Linear Independence of a Subset of $M_{2\times 2}$

Question

Determine if the following subset of $M_{2\times 2}$ is linearly independent: $$U = \left\lbrace\begin {bmatrix} 1&1 \\\\ 0&1 \ \end{bmatrix}, \begin {bmatrix} 1&0 \\\\ 1&1 \ \end{bmatrix}, \begin {bmatrix} 1&1 \\\\ 1&1 \ \end{bmatrix} \right\rbrace$$

I am approaching the problem with: $$ c_1 \begin {bmatrix} 1&1 \\\\ 0&1 \ \end{bmatrix} + c_2 \begin {bmatrix} 1&0 \\\\ 1&1 \ \end{bmatrix} + c_3 \begin {bmatrix} 1&1 \\\\ 1&1 \ \end{bmatrix} = 0$$ from the above I get: $$c_1+c_2+c_3=0 \\ c_1 + c_3=0 \\ c_2+c_3=0 \\ c_1+c_2+c_3=0$$ but when I solve it, I end up with an inconsistent system. Doesn't this mean the system is neither LI or LD? Or did I mess up somewhere.

Answer 1

Substituting the 2nd equation into the 1st gives $c_2=0$. Substituting into the 3rd equation gives $c_3=0$, and then the 1st equation gives $c_1=0$. Therefore the vectors are linearly independent.