Question: Consider the LU factorization of a matrix A without pivoting (assuming it exists): A = LU, and its QR factorization: A = QR. Assume that

Consider the LU factorization of a matrix A without pivoting (assuming it exists): A = LU, and its QR factorization: A = QR. Assume that the entries of A are random, that is they are normally distributed random variables. Show that the last column of Q is a random unit vector (that is a vector with uniform random orientation on the unit ball). Describe a relationship between the last row of L1 and the last column of Q.

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