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In: Advanced Math

9. Let L : R 10 → R 10 be a linear function such that the...

9. Let L : R 10 → R 10 be a linear function such that the composition L ◦ L is the zero map; that is, (L ◦ L)(x) = L L(x) = ~0 for all ~x ∈ R 10 . (a) Show that every vector v in the range of L belongs to the the kernel ker(L) of L. (b) Is it possible that ker(L) and Range(L) both have dimension bigger than 5? Carefully justify your answer. (c) Let A be a representing matrix for L. Show that the rank of A does not exceed 5

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