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## Homework Statement

Let ##L: R^{2} → R^{2}## be a linear map such that ##L ≠ O## but## L^{2} = L \circ L = O.##

Show that there exists a basis {##A##, ##B##} of ##R^{2}## such that:

##L(A) = B## and ##L(B) = O.##

## The Attempt at a Solution

Here's the solution my book provides :

Well I have two questions:

1.Why do they say that ##aA+bB=O##?. I mean I don't understand the solution from that point until the end (Why the solutions ##a=0## and ##b=0## are enough to prove the existence of that basis??May someone please explain??

Thanks in advance . Any help would be appreciated