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For example this matrix set here.

$$A_1=\begin{bmatrix}1&1\\0&1\end{bmatrix}, A_2=\begin{bmatrix}2&-1\\0&1\end{bmatrix}, A_3=\begin{bmatrix}3&6\\0&4\end{bmatrix}$$

How can I determine this set of matrices are linearly independent or not? Like I know how to calculate linear independence on sets like: $$v_1=(2, −1, 5) \ v_2=(1, 3, −4) \ v_3=(−3, −9, 12)$$ but I dont know how to do that thing on matrices.

thank you.

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  • $\begingroup$ thanks for the edit alain. $\endgroup$
    – damacaner
    Nov 17, 2022 at 18:09
  • $\begingroup$ Welcome to math SE. I took the liberty to write the matrix for you. It is better to use mathjax instead of picture, it is easier to search. $\endgroup$
    – Alain Remillard
    Nov 17, 2022 at 18:10
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    $\begingroup$ Welcome. $\begin{bmatrix}3&6\\0&4\end{bmatrix}$, it's just like $(3,6,0,4)$. $\endgroup$
    – Stéphane Jaouen
    Nov 17, 2022 at 18:12
  • $\begingroup$ thank you a lot dudes! $\endgroup$
    – damacaner
    Nov 17, 2022 at 18:14

1 Answer 1

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The way you do for vectors in $\mathbb{R}^3$, you have to treat the matrices as vectors. So, set up the equation $$c_1A_1+c_2A_2+c_3A_3=\begin{bmatrix}0&0\\0&0\end{bmatrix}$$ To get $$\begin{bmatrix}c_1+2c_2+3c_3 &c_1-c_2+6c_3\\0&c_1+c_2+4c_3\end{bmatrix}=\begin{bmatrix}0&0\\0&0\end{bmatrix}$$

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