You are given the set S = {v_1..., v_5} from R^4 and the vector b = (-2,1, -2,2)

Question

You are given the set S = {v_1..., v_5} from R^4 and the vector b = (-2,1, -2,2). The associated augmented matrix is shown below, along with its RREF: Find all solutions x = (x_1 x_2, x_3, x_4, x_5) to the system. Be sure to clearly indicate which are the leading variables and which are the free variables. Express b as a linear combination of the vectors from S in the simplest way possible. Show that your combination is correct. Express b as a linear combination of the vectors with no x_1 = 0. Show that your combination is correct.

Explanation / Answer

A) Using RREF matrix, we get the system of original equations reduced to following:

x1 + 3x3 + x4 = 2

x2 + 2x3 - x4 = -1

x5 = -1

The leading variables correpesponds to the the column that has first non zero entry as 1....And as we see the matrix , the leading variables are x1, x2 and x5 and free variables are x3 and x4

Hence take x3 = t, x4 = s

x1 + 3t + s = 2 => x1 = 2-3t-s

x2 +2t -s = -1 => x2 = -1-2t+s

(x1, x2, x3, x4, x5) = (2-3t-s, 1-2t+s, s, t, -1)

These are all solutions of given systme

B) (-2,1,-2,2) = x1(1,2,2,3)+x2(2,1,2,1)+x3(7,8,10,11)+x4(-1,1,0,2)+x5(2,2,4,3)

(-2,1,-2,2) = (2,1,0,0,-1)+t(-3,-2,0,1,0)+s(-1,1,1,0,0)

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