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Find a basis of space of 2×2 matrices $\{A_1,A_2,A_3,A_4\} $such that $ A_i^2=A_i$ for all i. I know how to find basis but I am unable to find such a property for this. Can you help me? Two matrices can be :

[1,0,  and [0,0
 0,0].        0,1]
                            ```
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  • $\begingroup$ Have you tried brute force? Take an arbitrary matrix $A$ and square it element by element. $\endgroup$ – amd Apr 25 at 1:43
  • $\begingroup$ Sorry I didn't understand $\endgroup$ – Tojrah Apr 25 at 1:46
  • $\begingroup$ You seem to be assuming that you need $4$ elements for this basis. Why do you assume this? $\endgroup$ – The Count Apr 25 at 1:49
  • $\begingroup$ Also, the question phrasing is strange. Do you mean "find a basis for the subspace of $2\times 2$ matrices made up of all matrices $A$ such that $A^2=A$? Or even better, "Find a basis for the subspace $\{A|A^2=A\}$ of $2\times 2$ matrices". Because I doubt that is even a subspace. $\endgroup$ – The Count Apr 25 at 1:52
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    $\begingroup$ OH OH OH I get it. You need just a basis for the space, and each member of the basis needs to have the property that it is its own square. I see. Well, what is a basis for $V$ that you know? $\endgroup$ – The Count Apr 25 at 1:54
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Try adding $[1,1,0,0]$ and $[0,0,1,1]$.

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  • $\begingroup$ Yeah, got it... $\endgroup$ – Tojrah Apr 25 at 2:02

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