01:640:135 Midterm: 01:640:135 Test Fall 2016

Consider the following Gauss-Jordan reduction: [-5 0 1 5 -4 -1 1 0 0]_ A rightarrow [-5 0 1 0 -4 0 1 0 0]_ E_1A rightarrow [0 0 1 0 -4 0 1 0 0]_ E_2E_1A rightarrow [1 0 0 0 -4 0 0 0 1]_ E_3E_2E_1A rightarrow [1 0 0 0 1 0 0 0 1]_ E_4E_3E_2E_1 A = I Find E_1 = [], E_2 = [], E_3 = [], E_4 = []. Write A as a product A = E_1^-1 E_2^-1 E_3^-1 E_4^-1 of elementary matrices: [-5 0 1 5 -4 -1 1 0 0] =

Find a least - squares solution of Ax = b for A = [1 1 1 1 1 1 1 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 1], b = [- 3 -1 0 2 5 1] The general least - squares solution of Ax = b has the form X = [(1)] + x4 [(2)]

Termine its reduced row echelon form. Record the row operations you perform, using the notation for elementary row operations. (a) A = [1 0 0 0 0 1 0 0 -3 1 1 0 2 1 2 0] (b) A = [1 0 0 3 1 0 0 0 1 2 1 -1 4 0 0]