Linear algebra-Basis of a linear map

AI Thread Summary
The discussion revolves around a linear map L from R² to R², where L is not the zero map but L² equals the zero map. The task is to demonstrate that there exists a basis {A, B} for R² such that L(A) equals B and L(B) equals the zero vector. Participants seek clarification on the implications of linear independence and why the conditions aA + bB = 0 lead to a and b both being zero. It is emphasized that since L is a linear map, it follows that L(aA + bB) equals aL(A) + bL(B), reinforcing the concept of linearity in the context of vector spaces. Understanding these properties is crucial for establishing the required basis.
manuel325
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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 :
problem.JPG

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 :smile:. Any help would be appreciated
 
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The solution says IF aA+bB=0, THEN a=b=0. That is what it means for the vectors A and B to be linearly independent. Vectors in a basis must be linearly independent.
 
krome said:
The solution says IF aA+bB=0, THEN a=b=0. That is what it means for the vectors A and B to be linearly independent. Vectors in a basis must be linearly independent.
Thanks but why ##O=L(aA+bB)=aL(A)##?? could you please explain what they do there, please??
 
manuel325 said:
Thanks but why ##O=L(aA+bB)=aL(A)##?? could you please explain what they do there, please??

L is a linear map, which means L(aA+bB) = aL(A) + bL(B).
 
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krome said:
L is a linear map, which means L(aA+bB) = aL(A) + bL(B).

Thanks :smile:
 
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