I know how to prove whether or not vectors are linearly independent, but can't apply the same thing to matrices it seems. Given three 2x2 matrices, for example:
$$A = \begin {bmatrix} -1&1 \\\\ -1&1 \ \end{bmatrix}$$ $$B = \begin {bmatrix} 1&1 \\\\ -1&-1 \ \end{bmatrix}$$ $$C = \begin {bmatrix} -1&1 \\\\ 1&-1 \ \end{bmatrix}$$
I want to test whether or not these are linearly dependent. So with vectors I would do something like:
$$ c_1A + c_2B + c_3C = 0$$
Where the $c_i$'s are some scalar constants, and prove that the only solution of that is when $$c_1 = c_2 = c_3 = 0$$
So how do I go about solving this:
$$ c_1 \begin {bmatrix} -1&1 \\\\ -1&1 \ \end{bmatrix} + c_2 \begin {bmatrix} 1&1 \\\\ -1&-1 \ \end{bmatrix} + c_3 \begin {bmatrix} -1&1 \\\\ 1&-1 \ \end{bmatrix} = 0$$
Or I am going about this completely the wrong way?
Any help would be hugely appreciated.