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A linear transformation L:V->V is said to be idempotent if L dot L = L. If L is idempotent, show that there exists a basis S={a1,a2,...,an} for V such that L(ai)=ai for i= 1,2,...,r and L(aj) = 0v for j= r+1,...,n, where r= p(L). Describe the matrix representing L with respect to the basis S.

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If L is idempotent, show that there exists a basis S={a1,a2,...,an} for V

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To show that this we first find a basis for the image of L as well as its kernel. Let's assume that u1, u2, ..., ur are r linearly independent vectors in V( as r=p(L)) which span the image of L. Then because these are linearly independent vectors which form a basis for the image of L we must have that L(u1), L(u2), ..., L(ur) are also linearly independent. We call L(u1)=a1, L(u2)=a2, ..., L(ur)=ar.
Then we ...

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