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why is it that when findinf the basis for a set of vectors you row reduce...

why is it that when findinf the basis for a set of vectors you row reduce the augmented matrix with zero and then find the pivot columns ans the set of correspondingvectors in the original matrix form the basis, however whej youre finding a basis for the eigenspace, and you do A-lambda*I, why dont you do the same procedure with that matrix?
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