 # 2.) Let n be an odd positive integer. Determine whether there exis...

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2.) Let n be an odd positive integer. Determine whether there exists an n x n real matrix A such that (A^ 2) + = o, where I is the nx n identity matrix and O is the n xn zero matrix. If such a matrix A exists, find an example. If not, prove that there is no such A. How about when n is an even positive number? 3.) For two subspaces U and W of a vector spaces V, we define their sum as U+ W= {v :v=u+w:u e U, w e W}. It is well-known that dim(U + W) S dim(U) + dim(W). Eventually, using the above inequality, show that if A, B are m x n matrices, then rank(A + B) < rank(A) + rank(B). 4.) Let V be the subspace of (R^4) defined by the equation x1 x2 + 2x3 + 6x4 =0. Find a linear transformation T from (R^3) to (R^4) such that the null space Null(T) = {0} and the range Range(T) = V. Describe T by its matrix A. 5.) Let V be the vector space of 2x: 2 matrices. Suppose that the linear transformation from V to V is given by T(A) = 123;57] A A[23;57]. Prove or disprove that the linear transformation T: V - V is an isomorphism.

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