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Let $V$ be the space of $2\times 2$ matrices over $F.$ Find a basis $\{A_1,A_2,A_3,A_4,\}$ for $V$ such that $A_j^2=A_j,$ for each $j.$

Let $X=\begin{bmatrix}x_1 & x_2\\x_3&x_4\end{bmatrix}.$ We want $X^2=\begin{bmatrix}x_1 & x_2\\x_3&x_4\end{bmatrix}\begin{bmatrix}x_1 & x_2\\x_3&x_4\end{bmatrix}=\begin{bmatrix}x_2x_3+x_1^2 & x_2(x_1+x_4)\\x_3(x_1+x_4)&x_2x_3+x_4^2\end{bmatrix}=\begin{bmatrix}x_1 & x_2\\x_3&x_4\end{bmatrix}.$

Looking at the equations I wrote these vectors $\{\begin{bmatrix}1 & 0\\0&0\end{bmatrix},\begin{bmatrix}0 &0\\0&1\end{bmatrix},\begin{bmatrix}0 & 0\\1&1\end{bmatrix},\begin{bmatrix}1 & 1\\0&0\end{bmatrix}\}$ as the basis.

I want to arrive at the general vectors just by solving those equations rather than guessing some particular solutions. Can you show me how to go about obtaining the general solutions?

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  • $\begingroup$ What do you mean by general solution? $\endgroup$ – JJR May 15 '17 at 15:55
  • $\begingroup$ Well, I meant to say that some solution matrices like $\begin{bmatrix}a & 0\\0&0\end{bmatrix}.$ $\endgroup$ – Bijesh K.S May 15 '17 at 16:14
  • $\begingroup$ which would hold in general for any constants that I plug in $\endgroup$ – Bijesh K.S May 15 '17 at 16:15
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    $\begingroup$ See the Wikipedia article on idempotent matrices for a discussion on the real $2 \times 2$ case. Since the field $F$ is arbitrary in your question, the only elements of the field that you can explicitly write down are $0$ and $1$, so it is best to search for an example using just these elements, as you have done. $\endgroup$ – Brahadeesh Jan 12 '18 at 14:58
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From the comments above.


See the Wikipedia article on idempotent matrices for a discussion on the real $2 \times 2$ case. Since the field $F$ is arbitrary in your question, the only elements of the field that you can explicitly write down are $0$ and $1$, so it is best to search for an example using just these elements, as you have done.

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