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 May 20 awarded Famous Question Mar 21 awarded Notable Question Nov 20 awarded Popular Question Dec 9 awarded Scholar Dec 9 accepted Are three matrices linearly independent and form a basis of $M_2(\mathbb R)$? Dec 9 comment Are three matrices linearly independent and form a basis of $M_2(\mathbb R)$? Ok thanks, that's all I really needed to know I suppose. I solved the question like this: $$\operatorname{rref}\left[\begin{array}{ccc|c} -1&1&-1&0 \\ 1&1&1&0 \\ -1&-1&1&0 \\ 1&-1&-1&0 \end{array}\right] = \left[\begin{array}{ccc|c} 1&0&0&0 \\ 0&1&0&0 \\ 0&0&1&0 \\ 0&0&0&0 \end{array}\right]$$ so the matrices are linearly independent. Assuming that's right, my question has been answered. Dec 9 comment Are three matrices linearly independent and form a basis of $M_2(\mathbb R)$? So I can rewrite the matrices like this: $$A = (-1, 1, -1, 1)$$ and so on for the others. How does that simplify the problem? I will get: $$c_1(-1, 1, -1, 1) + c_2(1, 1, -1, -1) + c_3(-1, 1, 1, -1) = 0$$ Will that give me four equations? Dec 9 awarded Student Dec 9 asked Are three matrices linearly independent and form a basis of $M_2(\mathbb R)$?