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3（b)（c)（d)

3（b)（c)（d)

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
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Answer3（b)（c)（d)

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
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