Problem 3. Let 1 1 2 A = 10 022 12 (a) Find an ordered basis for ran(A) (b) Perform the Gram-Schmidt algorithm on the basis you got in (a) to get an orthonormal basis for ran(A) (c) Find a basis for ker(A) (d) Extend the basis you found in (a) to a basis B for R³ (e) Find Pe→B. Let A be as in problem 3. (a) Compute the matrix of proj ran(A), that is, find the matrix M so that Mã = projran(A)(x) for all x € R³. (b) For each of the following b; compute projran (A) (bi) : 51 [2] = 1 18 = 2 ხვ = Ба = 322 (c) For each 5; from (b), let ỗ¿ = projran(A) (ō;), each ĥ; has Ĝi Є ran(A) and so there is a solution to Ax = bi. Use ker(A) and the change of basis matrix you found in problem 3 to find full solution sets to for each i = 1, 2, 3, 4. Axi = bi
Problem 3. Let 1 1 2 A = 10 022 12 (a) Find an ordered basis for ran(A) (b) Perform the Gram-Schmidt algorithm on the basis you got in (a) to get an orthonormal basis for ran(A) (c) Find a basis for ker(A) (d) Extend the basis you found in (a) to a basis B for R³ (e) Find Pe→B. Let A be as in problem 3. (a) Compute the matrix of proj ran(A), that is, find the matrix M so that Mã = projran(A)(x) for all x € R³. (b) For each of the following b; compute projran (A) (bi) : 51 [2] = 1 18 = 2 ხვ = Ба = 322 (c) For each 5; from (b), let ỗ¿ = projran(A) (ō;), each ĥ; has Ĝi Є ran(A) and so there is a solution to Ax = bi. Use ker(A) and the change of basis matrix you found in problem 3 to find full solution sets to for each i = 1, 2, 3, 4. Axi = bi
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter2: Systems Of Linear Equations
Section2.2: Direct Methods For Solving Linear Systems
Problem 3CEXP
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