Homework Help is Here – Start Your Trial Now!
Math
Advanced Math
Main Thm Let F=Rov ( three on LS solutions Given conditions for a (a) s LS ibis 15 solution a b' = orth prof of b IC) SIS is a any AM (F) SEF" Oh vector solution of of an (exact) solution of and be the following Fm, equivalent: the system Ax=b the system Ax=b', where col(A) + col(A) = R(T) the are system A*Ax= Ab

Main Thm Let F=Rov ( three on LS solutions Given conditions for a (a) s LS ibis 15 solution a b' = orth prof of b IC) SIS is a any AM (F) SEF" Oh vector solution of of an (exact) solution of and be the following Fm, equivalent: the system Ax=b the system Ax=b', where col(A) + col(A) = R(T) the are system A*Ax= Ab

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
See similar textbooks
icon
Related questions
Question

Please don't answer 3(a)

3. Let F R or C, WC F be an m-dimensional subspace, and Pw: FnF be the orthogonal projection onto W. Choose a basis {w₁,..., wm} for W, and let A be the nxm matrix whose ith column is w; for i = 1, ..., m. Let v € Fn. Since Pw(v) E W and {w₁,..., wm} is a basis for W, we can write Pw (v) = a₁w₁ + + amwm for some unique a₁,..., am € F. (a) Use the main theorem on least squares solutions to show that the set of all least squares solutions to Ax = v = Pw (v) A*v. = the set of all solutions to Ax= = the set of all solutions to A*Ax = (b) Use the first equality in (a) to show that the system Ax = v has a unique least a1 ·0 Am squares solution, which is (c) Now use the second equality in (a) and the result of (b) to show that the matrix A* A is invertible, and (A*A)-¹A* v= (0) Am (d) (Formula for the orthogonal projection matrix). Use the result of (c) to show that the matrix representation of Pw relative to the standard basis for Fn is A(A*A)−¹A*.
Transcribed Image Text:3. Let F R or C, WC F be an m-dimensional subspace, and Pw: FnF be the orthogonal projection onto W. Choose a basis {w₁,..., wm} for W, and let A be the nxm matrix whose ith column is w; for i = 1, ..., m. Let v € Fn. Since Pw(v) E W and {w₁,..., wm} is a basis for W, we can write Pw (v) = a₁w₁ + + amwm for some unique a₁,..., am € F. (a) Use the main theorem on least squares solutions to show that the set of all least squares solutions to Ax = v = Pw (v) A*v. = the set of all solutions to Ax= = the set of all solutions to A*Ax = (b) Use the first equality in (a) to show that the system Ax = v has a unique least a1 ·0 Am squares solution, which is (c) Now use the second equality in (a) and the result of (b) to show that the matrix A* A is invertible, and (A*A)-¹A* v= (0) Am (d) (Formula for the orthogonal projection matrix). Use the result of (c) to show that the matrix representation of Pw relative to the standard basis for Fn is A(A*A)−¹A*.
Main Thm Let F=R ov ( Given three conditions for a on LS (a) s ibi s is b' = orth 1C) S is is a solutions any a A = Mmx (F) LS solution pros of b an (exact) solution of vector solution of of Oh SEF" and be FM, the following equivalent: the are the system Ax=b the system Ax=b', where cal(A) & col(A) = R(T) system A*Ax=A* b
Transcribed Image Text:Main Thm Let F=R ov ( Given three conditions for a on LS (a) s ibi s is b' = orth 1C) S is is a solutions any a A = Mmx (F) LS solution pros of b an (exact) solution of vector solution of of Oh SEF" and be FM, the following equivalent: the are the system Ax=b the system Ax=b', where cal(A) & col(A) = R(T) system A*Ax=A* b
Expert Solution
steps

Step by step

Solved in 5 steps with 4 images

SEE SOLUTIONCheck out a sample Q&A here
Blurred answer
Follow-up Questions
Read through expert solutions to related follow-up questions below.
Follow-up Question

Answer 3(b)

Solution
Bartleby Expert
SEE SOLUTION
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
Basic Technical Mathematics
Basic Technical Mathematics
Advanced Math
ISBN:
9780134437705
Author:
Washington
Publisher:
PEARSON
Topology
Topology
Advanced Math
ISBN:
9780134689517
Author:
Munkres, James R.
Publisher:
Pearson,
SEE MORE TEXTBOOKS