Class Notes (834,337)
Mathematics (1,171)
MATH125 (103)
Nikita (37)
Lecture 35

# MATH125 Lecture 35: 35

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School
Department
Mathematics
Course
MATH125
Professor
Nikita
Semester
Winter

Description
▯▯▯▯▯▯▯ ▯▯▯ ▯▯▯▯▯ ▯▯ ▯▯▯▯▯ 3 Example. ▯▯▯▯ ▯▯ ▯▯▯▯▯▯▯▯▯▯ ▯▯▯▯▯ ▯▯▯ ▯▯▯ ▯▯▯▯▯▯▯▯ W ▯▯ R ▯▯▯▯▯ ▯▯ ▯ ▯ ▯ ▯ ▯ x ▯ ▯ ▯ W ▯ ▯ y ▯ x ▯ y ▯ ▯z ▯▯▯ z Solution. W ▯▯▯▯▯▯▯▯ ▯▯ ▯▯▯▯▯▯▯ ▯▯ ▯▯▯▯▯▯▯▯ y ▯ ▯z ▯ ▯▯ ▯ y ▯ ▯ ▯ ▯ ▯ ▯ ▯ y ▯ z z ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯▯ ▯▯ ▯▯▯▯▯▯▯ ▯▯▯▯ u ▯▯ ▯▯▯ v ▯ ▯ ▯▯ ▯▯▯▯ ▯ ▯▯▯▯▯ ▯▯ W▯ ▯▯▯ ▯▯▯▯ ▯ ▯ ▯▯▯ ▯▯▯ ▯▯▯▯▯▯▯▯▯▯▯ ▯▯ ▯▯ﬃ▯▯▯ ▯▯ ▯▯▯ ▯▯▯▯▯▯▯ ▯▯▯▯▯▯▯ ▯▯▯▯▯▯ ▯▯W ▯▯▯▯ ▯▯ ▯▯▯▯▯▯▯▯▯▯ ▯▯ ▯▯▯▯▯▯ ▯▯▯ ▯▯ ▯▯▯▯▯▯ ▯ ▯ x ▯▯▯▯▯▯▯ w ▯ ▯ y ▯▯ ▯ ▯▯▯▯▯▯ ▯▯ W ▯▯▯▯ ▯▯ ▯▯▯▯▯▯▯▯▯▯ ▯▯ u▯ ▯▯▯▯ z x▯y ▯▯z ▯ ▯ ▯▯▯▯▯ w ▯▯ ▯▯ W▯ ▯▯▯▯▯ u▯w ▯ ▯▯ ▯▯ ▯▯▯▯ ▯▯▯▯ x▯y ▯ ▯▯ ▯▯▯▯▯▯▯ ▯▯▯ ▯▯▯▯▯▯ ▯▯▯▯▯▯ ▯▯ ▯▯▯▯▯ ▯▯▯ ▯▯▯▯▯▯▯▯▯▯ ▯▯ ▯▯▯ ▯▯▯▯ x ▯ ▯z ▯▯▯ y ▯ z▯ ▯▯▯▯▯ ▯▯▯ ▯▯▯▯▯▯▯ ▯▯▯▯▯▯ w ▯▯ ▯▯▯ ▯▯▯▯ ▯ ▯ ▯z w ▯ ▯ z ▯ z ▯ ▯ ▯▯ ▯▯▯▯ ▯▯▯ ▯▯ ▯▯ ▯▯▯▯▯▯▯▯ ▯▯ ▯▯▯▯ w ▯▯▯ ▯ ▯▯▯▯▯ ▯▯▯W ▯ ▯▯ ▯u;w▯ ▯ ▯▯ ▯▯ ▯▯▯▯▯▯▯▯▯▯ ▯▯▯▯▯ ▯▯▯ W▯ ▯▯▯▯▯▯▯ ▯▯▯▯▯▯▯▯▯ ▯▯ ▯▯▯▯▯▯▯ ▯▯▯▯ ▯▯ ▯▯▯▯▯▯▯▯▯▯ ▯▯▯▯▯ ▯▯ ▯▯▯▯ ▯▯▯ ▯▯▯▯▯▯▯▯▯▯▯ ▯▯ ▯ ▯▯▯▯▯▯ ▯▯▯▯ ▯▯▯▯▯▯▯ ▯▯ ▯▯▯▯ ▯ ▯▯▯▯▯ ▯▯▯ ▯▯▯▯ ▯▯ ▯▯▯▯ ▯▯▯▯▯ ▯▯▯▯▯▯▯ ▯▯▯▯▯ ▯▯ ▯ ▯▯▯▯▯▯▯ ▯▯▯ ▯▯▯▯▯ ▯▯▯▯▯▯▯▯▯▯▯▯ ▯▯ ▯▯▯ ▯▯▯▯▯▯▯▯▯ ▯▯▯▯▯▯▯ ▯▯▯▯▯▯▯▯▯▯▯▯ Theorem. Let ▯v ;:::;v ▯ be an orthogonal basis for a subspace W of n 1 k R and let w be any vector in W. Then the coordinates1c ;::k;c of the vector w with respect to the basis are given by w ▯ v ci▯ i for i ▯ ▯;:::;k: vi▯ i Proof. ▯▯▯ ▯▯▯▯▯▯▯▯▯▯▯ ▯▯▯▯▯▯▯ ▯▯▯ ▯▯▯▯▯▯▯▯ w ▯ c v ▯ ▯▯▯ ▯ c v : 1 1 k k ▯▯▯▯▯▯ ▯▯▯ ▯▯▯ ▯▯▯▯▯▯▯ ▯▯▯▯iv ▯ ▯▯ ▯▯▯ ▯▯▯▯iw ▯ivi▯ i ▯v ▯ ▯ ▯: ▯ ▯ ▯ ▯▯▯ Example. ▯▯▯▯ ▯▯▯ ▯▯▯▯▯▯▯▯▯▯▯ ▯▯ w ▯ ▯▯▯▯ ▯▯▯▯▯▯▯ ▯▯ ▯▯▯ ▯▯▯▯▯ ▯ ▯▯▯▯▯▯▯ ▯▯ ▯▯ ▯▯ ▯▯▯ ▯▯▯▯▯▯▯▯ ▯▯▯▯▯▯▯▯▯ ▯▯▯▯▯▯▯▯▯▯ ▯▯▯▯▯ ▯▯ R ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ v ▯ ▯ ▯ ▯ ;v ▯ ▯▯▯ ;v ▯ ▯ ▯▯▯ 1 2 3 ▯▯ ▯ ▯ Solution. ▯▯▯ ▯▯▯▯▯▯▯▯▯▯▯ ▯▯▯ w ▯ 1 ▯ w ▯ 2 ▯ w ▯ 3 ▯ c1▯ ▯ ; 2 ▯ ▯ ; c3▯ ▯ v1▯ 1 ▯ v2▯ 2 ▯ v3▯ 3 ▯ ▯▯▯▯▯▯▯ ▯▯▯ ▯▯▯▯▯▯▯▯▯ ▯▯ ▯▯▯▯ ▯▯▯▯▯▯▯ ▯▯▯▯ ▯▯▯ ▯▯▯▯ ▯▯▯▯▯▯▯▯ ▯▯ ▯▯▯ ▯▯▯▯▯ ▯▯▯▯▯▯▯▯▯▯▯ ▯▯▯▯▯▯▯▯ ▯▯▯ ▯▯▯ ▯▯▯▯▯▯ ▯▯▯▯▯ ▯▯ ▯▯▯▯▯▯▯▯▯▯ ▯▯▯ ▯▯▯▯▯ ▯▯ ▯▯▯▯▯▯▯▯▯▯ ▯▯▯▯▯▯ ▯▯ ▯▯▯▯▯ ▯▯ ▯▯▯ ▯▯▯▯▯▯▯▯▯ ▯▯ ▯▯▯▯ ▯▯▯▯▯▯▯▯ ▯▯▯ ▯▯▯▯▯ ▯▯▯▯▯▯▯▯ ▯▯ ▯▯▯ ▯▯▯▯▯▯▯▯ ▯▯▯▯▯ ▯▯ R ▯▯ ▯▯▯▯ ▯▯▯▯ ▯▯▯▯▯▯▯▯ ▯▯▯▯▯ ▯▯▯▯▯▯ ▯▯ ▯ ▯▯▯▯ ▯▯▯▯ ▯▯▯▯ ▯▯▯▯▯▯▯▯▯ ▯▯▯▯ ▯▯▯▯▯▯▯▯ ▯▯▯▯ ▯▯▯▯▯▯▯▯▯▯▯▯▯▯ ▯▯ ▯▯▯▯ ▯▯▯ ▯▯▯▯▯▯▯▯▯ ▯▯▯▯▯▯▯▯▯▯ De▯nition. ▯ ▯▯▯ ▯▯ ▯▯▯▯▯▯▯ ▯▯ R ▯▯ ▯▯ orthonormal set ▯▯ ▯▯ ▯▯ ▯▯ ▯▯▯▯▯▯▯▯▯▯ ▯▯▯ ▯▯ ▯▯▯▯ ▯▯▯▯▯▯▯▯ ▯▯ orthonormal basis ▯▯▯ ▯ ▯▯▯▯▯▯▯▯ W ▯▯ R ▯▯ ▯ ▯▯▯▯▯ ▯▯ W ▯▯▯▯ ▯▯ ▯▯ ▯▯▯▯▯▯▯▯▯▯▯ ▯▯▯▯ Remark. ▯▯ ▯▯▯▯▯ ▯▯▯▯ ▯ ▯▯▯ ▯▯ ▯▯▯▯▯1▯ ▯q k:::;q ▯ ▯▯ ▯▯▯▯▯▯▯▯▯
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