Question

(Linear Algebra)

Please explain how to get to the answers step by step.

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The vector x is in a subspace H with a basis ß lb1, b2). Find the ß-coordinate vector of x. 10 -3 25 Objective: (2.9) Find Beta-Coordinate Vector of x Determine the rank of the matrix. 1-2 3 -5 16) 2 -4 8 -6 3 6-915 Objective: (2.9) Determine Rank of Matrix

Answer 1

We want: X = [p q] where p and q are the coordinates expressing x in the terms of b_1 & b_2 so, x = p b_1 + q b_2 Therefore [10 11 - 25] = p [2 - 2 4] + q [6 1 - 3] We get, 2 p + 6 q = 10 - 2 p + q = 11 4 p - 39 = -25 solving system of linear equation We get, p = -4 q = 3 so, x = [- 4 3] as a beta - coordinate vector of x Given matrix is A = [1 2 - 3, - 2 - 4 6, 3 8 - 9, - 5 - 6 15] first We have finding row echelon matrix R_2 rightarrow R_2 - 2 R_1 R_3 rightarrow R_3 + 2 R_1 [1 0 0, - 2 0 0, 3 2 0, - 5 4 0] R_2 rightarrow (1/2) R_2 [1 0 0, - 2 0 0, 3 1 0, - 5 2 0]