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Lecture 26

MATH125 Lecture 26: 26

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Department
Mathematics
Course
MATH125
Professor
Nikita
Semester
Winter

Description
▯▯▯▯▯▯▯ ▯▯▯ ▯▯▯▯▯ ▯▯▯ ▯▯▯▯▯ ▯▯▯ ▯▯▯▯▯ ▯▯▯▯▯▯▯ ▯▯▯▯▯ ▯▯▯▯ ▯▯ ▯▯▯▯▯▯▯ ▯▯▯▯ ▯▯▯ ▯▯▯ ▯▯ ▯▯▯ ▯▯▯▯▯▯ ▯▯▯▯▯▯▯ ▯▯▯▯▯▯▯▯▯▯▯▯▯ ▯▯ ▯▯ ▯▯▯▯▯▯▯▯▯▯ ▯ ▯▯ ▯ ▯▯▯▯▯▯ ▯▯▯▯▯▯ A ▯▯ ▯▯▯▯ n▯n ▯▯ ▯▯▯▯ ▯▯▯ ▯▯▯ ▯▯ ▯▯▯ ▯▯▯▯▯▯▯ ▯▯▯▯▯▯▯ ▯▯ ▯▯▯ ▯▯▯▯ ▯▯▯▯▯ ▯▯ ▯▯▯ ▯▯▯▯▯▯ A ▯ ▯In▯ De▯nition. Let A be an n ▯ n matrix and let ▯ be an eigenvalue of A. The collection of all eigenvectors of A corresponding to ▯ together with the zero vector is called the eigenspace of ▯ and is de▯oted by E . Example. ▯▯▯▯ ▯▯▯▯ ▯ ▯ ▯ ▯▯ ▯▯ ▯▯▯▯▯▯▯▯▯▯ ▯▯ ▯ ▯ ▯ ▯ ▯▯ A ▯ ▯ ▯▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯▯▯ ▯▯▯ ▯ ▯▯▯▯▯ ▯▯▯ ▯▯▯ ▯▯▯▯▯▯▯▯6▯ E ▯ Solution. ▯▯ ▯▯▯▯ ▯▯ ▯▯▯▯▯▯▯ ▯▯▯ ▯▯▯▯ ▯▯▯▯▯ ▯▯ A ▯3▯I ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯▯ ▯ ▯ ▯▯ A ▯ ▯I3▯ ▯ ▯▯ ▯▯ ▯ ▯ ▯▯ ▯ ▯ ▯ ▯ ▯ : ▯ ▯ ▯▯ ▯ ▯ ▯ ▯▯ ▯▯▯ ▯▯▯▯ ▯▯▯▯▯ ▯▯▯ ▯▯▯ ▯▯▯▯ ▯▯▯▯▯▯▯▯▯ ▯▯▯▯▯▯▯▯ ▯▯▯ ▯▯▯▯▯▯▯3▯ ▯▯ A▯▯I ▯▯ ▯▯▯▯▯▯▯▯ ▯▯▯▯▯ ▯ ▯▯ ▯▯ ▯▯▯▯▯▯▯▯▯▯▯ ▯▯▯ ▯▯▯▯▯▯▯ ▯▯▯▯▯▯▯▯ ▯▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯x2▯ ▯x 3 ▯▯ ▯ ▯ x ▯ ▯ ▯ ▯ ▯ ▯ ▯ 2 ▯ x 2 ▯ x 3 : x3 ▯ ▯ ▯ ▯▯▯▯▯ ▯▯ 6 ▯▯ ▯ ▯ ▯ ▯ ▯▯ ▯ ▯ ▯ ▯ ; ▯ ▯ ▯ : ▯ ▯ ▯▯ ▯▯▯ ▯▯▯▯ ▯▯▯ ▯▯ ▯▯▯ ▯▯▯▯▯▯▯▯▯▯▯▯ ▯▯▯▯ ▯▯ ▯▯▯▯ ▯▯▯ ▯▯▯▯▯▯▯▯▯▯▯ ▯▯▯ ▯▯▯▯▯▯▯▯▯▯▯▯ ▯▯▯ ▯▯▯ ▯▯▯▯▯▯▯▯▯ ▯▯▯▯▯▯▯▯ ▯▯▯ ▯▯ ▯▯▯ ▯▯▯ ▯▯▯▯▯▯▯▯▯▯▯ ▯▯ ▯ ▯▯▯▯▯ ▯▯▯▯▯▯▯ ▯▯ ▯▯▯▯ ▯▯▯▯▯▯▯ ▯▯ ▯▯▯▯▯▯ ▯▯▯▯ ▯▯▯▯▯▯▯▯ ▯▯▯ ▯▯▯▯▯▯▯▯ ▯▯ ▯▯▯▯ ▯ ▯ ▯ ▯▯▯ ▯▯▯▯▯ ▯▯ ▯▯ ▯▯▯▯▯▯▯▯ ▯ ▯▯▯▯▯▯▯ ▯▯▯▯▯ ▯▯▯ A ▯▯ ▯ ▯▯▯▯▯▯ ▯▯ ▯▯▯▯ ▯ ▯ ▯▯ ▯ ▯▯▯▯▯▯ ▯ ▯▯ ▯▯ ▯▯▯▯▯▯▯▯▯▯ ▯▯ A ▯▯ ▯▯▯ ▯▯▯▯ ▯▯ ▯▯▯ ▯▯▯▯ ▯▯▯▯▯ ▯▯ A 2 ▯I ▯▯ ▯▯▯▯▯▯▯▯ ▯▯▯▯ ▯▯▯▯▯▯▯ ▯▯ ▯▯▯ ▯▯▯▯ ▯▯ ▯▯▯ ▯▯▯▯▯▯ A▯2I ▯▯ ▯▯▯ ▯▯▯▯▯▯▯▯▯▯ ▯▯▯▯▯▯▯▯▯▯ ▯▯▯ ▯▯▯ ▯▯▯▯▯▯▯▯▯▯ ▯▯▯▯▯▯ B ▯▯▯ ▯▯▯▯▯▯▯▯▯ nul▯B▯ ▯▯▯▯▯▯▯▯ ▯▯ ▯▯▯▯ ▯▯▯▯▯▯ ▯▯▯▯▯▯ ▯▯▯ ▯▯▯▯ ▯▯▯▯ ▯▯▯▯▯ ▯▯ ▯▯▯ ▯▯▯▯ ▯▯ ▯▯▯▯A2▯ ▯I ▯ ▯ ▯▯ ▯▯▯▯ ▯▯ ▯▯▯ ▯▯▯ ▯▯▯▯▯▯▯▯▯▯▯ ▯▯ A ▯▯ ▯▯▯▯ ▯▯▯▯▯ ▯▯▯ ▯▯▯▯▯▯▯▯ ▯▯▯▯A ▯ ▯I ▯ ▯ ▯: 2 Example. ▯▯▯▯ ▯▯▯ ▯▯▯▯▯▯▯▯▯▯▯ ▯▯▯ ▯▯▯ ▯▯▯▯▯▯▯▯▯▯▯▯▯ ▯▯▯▯▯▯▯▯▯▯▯▯ ▯▯ ▯▯▯ ▯▯▯▯▯▯ [ ] A ▯ ▯ ▯ : ▯ ▯ Solution. ▯▯ ▯▯▯▯ [ ] ▯ ▯ ▯ ▯ A ▯ ▯I2▯ : ▯ ▯ ▯ ▯ ▯▯▯▯▯ 2 ▯▯▯▯A ▯ ▯I2▯ ▯ ▯▯ ▯ ▯▯▯▯ ▯ ▯▯ ▯ ▯ ▯ ▯ ▯ ▯▯ ▯ ▯ ▯▯ ▯▯▯▯ ▯▯ ▯▯▯▯▯ ▯▯▯ ▯▯▯▯▯▯▯▯ ▯ ▯ ▯▯ ▯ ▯ ▯ ▯: ▯▯▯ ▯▯▯▯▯▯▯▯▯ ▯▯ ▯▯▯▯ ▯▯▯▯▯▯▯▯ ▯▯▯ ▯ ▯ ▯ ▯▯▯ ▯ ▯ ▯▯ ▯▯▯▯▯ ▯▯▯ ▯▯▯▯▯▯▯▯▯ ▯▯▯ ▯▯▯▯▯▯▯▯▯▯▯ ▯▯ A▯ ▯▯ ▯▯▯ ▯▯▯▯▯▯▯▯▯▯▯▯ ▯▯▯▯▯▯▯▯▯▯▯▯▯ ▯▯ ▯ ▯ ▯ ▯▯ ▯▯▯▯▯▯▯ ▯▯▯ ▯▯▯▯▯▯▯▯▯ ▯▯ A ▯ ▯I ▯ ▯▯ ▯▯▯ 2 [ ] [ ] ▯▯ ▯ ▯ ▯ ▯ ▯▯ ▯ ▯ ▯A ▯ ▯I2▯▯▯ ▯ ▯▯ ▯ ▯▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯▯▯▯ ▯▯▯▯▯ ▯▯ ▯▯▯▯▯▯▯▯ ▯▯▯▯ ▯▯▯ ▯▯▯▯▯▯▯▯▯▯▯▯ ▯▯▯▯▯▯▯▯▯▯▯▯▯ ▯▯ ▯ ▯ ▯ ▯▯▯ [ ] [ ] x2 ▯ ▯ x 2 : x2 ▯ ▯▯▯▯▯▯▯ ▯▯▯▯▯▯▯▯▯ ▯▯ ▯▯▯ ▯▯▯▯ ▯▯▯ ▯▯▯▯▯▯▯▯▯▯▯▯ ▯▯▯▯▯▯▯▯▯▯▯▯▯ ▯▯ ▯ ▯ ▯ ▯▯▯ ▯▯▯▯▯▯▯▯ ▯▯ [ ] ▯▯ : ▯ ▯▯▯▯▯ ▯▯▯▯▯▯▯▯▯▯▯▯ ▯▯▯▯▯▯ ▯▯▯▯ ▯ ▯▯▯▯▯▯ ▯▯ ▯▯▯▯ ▯ ▯ ▯ ▯▯ ▯▯▯▯▯▯▯▯▯▯ ▯▯ ▯▯▯ ▯▯▯▯ ▯▯ ▯▯▯ ▯▯▯▯▯▯▯▯▯▯▯ ▯▯ ▯▯▯▯▯▯▯▯ ▯▯ ▯▯▯▯▯▯ ▯▯▯▯ ▯▯▯▯▯▯ ▯▯▯▯ ▯▯ ▯▯▯▯▯▯ ▯▯▯▯▯▯▯▯ ▯▯ ▯▯▯▯ ▯
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