MATH 125 Lecture Notes - Lecture 12: Row Echelon Form, Augmented Matrix, Free Variables And Bound Variables

1. Consider the augmented matrix 2 4-8-2 28-24 -12 3 6 -13 47-42 -19 -1 -2 62-22 20 11 2 -3 0 -838 2 -63 1 2-2 -1 8-8 3 6 2 for a system of six equations in the variables r1, 2, r3, r4, r5 and re (a) Convert the above augmented matrix to echelon form by using the three elementary row operations we discussed in class (re- placing one row by the sum of itself and a multiple of another row, interchanging two rows, and multiplying a row by a nonzero number). At each stage, indicate the row operation you are using (b) Use the augmented matrix you obtained in the previous step to obtain a reduced echelon form for the augmented matrix. Again, peration you are using at each step. (c) Use the the reduced echelon form for the matrix that you obtained in the previous step to write the solution set for the corresponding system of equations. Indicate whether the system is inconsistent whether it has a unique solution or whether there are an infinite number of solutions. If there are an infinite number of solutions, an be any value) and the basic variables (meaning those variables that correspond to pivot columns). In addition, if there are an infinite number of solutions, use the solution set that you found to produce two indicate the free variables (meaning those that c individual 6-tuples that are solutions of the system.

Write these equations in matrix form and use row operations to get them to an equivalent of reduced row echelon form. What is the rank of the underlying matrix? v + w + z = 4 v + 2x + 2y + z = 7 v + w - y + z = 2 w - 2x + 2x = 5 v + 2x + y + z = 5 Find two homogeneous solutions which are not multiples of each other and a combination of these two solutions which contains 3 zeros. Check all of them the original equations. Why is it impossible to have a particular solution with more than two zeros in?

Use the row reduction algorithm to transform the matrix into echelon form or reduced echelon form as indicated. 3) Find the echelon form of the given matrix. [1 -3 2 4 -11 2 -2 9 5 3 -5 -1] A) [1 0 0 4 1 0 -2 3 27 3 4 17] B) [1 0 0 4 1 -6 -2 3 9 3 4 -7] C) [1 0 0 4 1 0 -2 3 27 3 4 0] D) [1 0 0 4 1 0 -2 3 15 3 4 -1] Solve the problem. 4) Let a_1 = [3 4 -4], a_2 = [-4 1 1], and b = [2 -10 6]. Determine whether b can be written as a linear combination of a_1 and a_2. In other words, determine whether we x_1 and x_2 exist, such that x_1 a_1 + x_2 a_2 = b. Determine the weights x_1 and x_2 if possible. A) x_1 = -1, x_2 = -3 B) x_1 = -2, x_2 = -2 C) x_1 = -2, x_2 = -1 D) No solution