# Linear independency at set of matrices.

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.

• thanks for the edit alain. Nov 17, 2022 at 18:09
• 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. Nov 17, 2022 at 18:10
• Welcome. $\begin{bmatrix}3&6\\0&4\end{bmatrix}$, it's just like $(3,6,0,4)$. Nov 17, 2022 at 18:12
• thank you a lot dudes! Nov 17, 2022 at 18:14

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 $$\begin{bmatrix}c_1+2c_2+3c_3 &c_1-c_2+6c_3\\0&c_1+c_2+4c_3\end{bmatrix}=\begin{bmatrix}0&0\\0&0\end{bmatrix}$$