# Finding a basis for vector space of matrices

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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• Have you tried brute force? Take an arbitrary matrix $A$ and square it element by element. – amd Apr 25 at 1:43
• Sorry I didn't understand – Tojrah Apr 25 at 1:46
• You seem to be assuming that you need $4$ elements for this basis. Why do you assume this? – The Count Apr 25 at 1:49
• 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. – The Count Apr 25 at 1:52
• 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? – The Count Apr 25 at 1:54

## 1 Answer

Try adding $$[1,1,0,0]$$ and $$[0,0,1,1]$$.

• Yeah, got it... – Tojrah Apr 25 at 2:02