Exercise 7: Let A = Rmx be a matrix such that its row are linearly independent...

 

Transcribed Image Text:

Exercise 7: Let A = Rmx be a matrix such that its row are linearly independent (as vectors in IR") and let b = Rm. such that the set H = {x R": Ax = b} Is nonempty. 1) Prove that H is a closed and convex subset of R". 2) Let a E R". By considering the optimization problem (7) (min||xa||2 and using the KKT theorem, prove that s.t. Ax = b PH (a) = a + AT (AAT)-(b Aa). Exercise 7: Let A = Rmx be a matrix such that its row are linearly independent (as vectors in IR") and let b = Rm. such that the set H = {x R": Ax = b} Is nonempty. 1) Prove that H is a closed and convex subset of R". 2) Let a E R". By considering the optimization problem (7) (min||xa||2 and using the KKT theorem, prove that s.t. Ax = b PH (a) = a + AT (AAT)-(b Aa).