Projection Matrix(a) Show that A is an n x n matrix and det(A) 0, then the projection matrix PA(ATA)-¹AT is the nxn identity matrix I (matrix with 1 on the diagonal and 0 outsideof the diagonal).(b) Consider the system equations r = 4 and r = 6. Obviously this system of equationsdoes not have an exact solution. Represent the system of equations as Av = b whereA = [1,1], v = [r], and b = [4,6]T. Find the projection matrix P. Show graphicallythe point b (the first coordinate on the z-axis and the second coordinate on the y-axis)and the point Pb. Show the set of points v € R2x1 such that Pu= v. What is the valueof a that solves Az = Pb?(c) Repeat the previous exercise for the system of equations r = 4 and 2x = 12. Keep inmind that you will obtain a different projection matrix. What is the value of a thatsolves Ax = Pb?(d) Find the approximate solution of the system of equations (using the projection matrix)[1 2]345 6] [C]1015

Answered: (a) Show that A is an n x n matrix and…

(a) Show that A is an n x n matrix and det(A) # 0, then the projection matrix P A(ATA)-¹A¹ is the nxn identity matrix I (matrix with 1 on the diagonal and 0 outside of the diagonal).

College Algebra

7th Edition

ISBN:9781305115545

Author:James Stewart, Lothar Redlin, Saleem Watson

Publisher:James Stewart, Lothar Redlin, Saleem Watson

Chapter6: Matrices And Determinants

Section: Chapter Questions

Problem 66E: Distribution of Cash An ATM at a bank in Qualicum Beach, British Columbia, dispenses $20 and $50...

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Question

Projection Matrix
(a) Show that A is an n x n matrix and det(A) 0, then the projection matrix P
A(ATA)-¹AT is the nxn identity matrix I (matrix with 1 on the diagonal and 0 outside
of the diagonal).
(b) Consider the system equations r = 4 and r = 6. Obviously this system of equations
does not have an exact solution. Represent the system of equations as Av = b where
A = [1,1], v = [r], and b = [4,6]T. Find the projection matrix P. Show graphically
the point b (the first coordinate on the z-axis and the second coordinate on the y-axis)
and the point Pb. Show the set of points v € R2x1 such that Pu= v. What is the value
of a that solves Az = Pb?
(c) Repeat the previous exercise for the system of equations r = 4 and 2x = 12. Keep in
mind that you will obtain a different projection matrix. What is the value of a that
solves Ax = Pb?
(d) Find the approximate solution of the system of equations (using the projection matrix)
[1 2]
34
5 6
] [C]
10
15

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