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Lecture 24

MATH125 Lecture 24: 24

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Department
Mathematics
Course
MATH125
Professor
Nikita
Semester
Winter

Description
▯▯▯▯▯▯▯ ▯▯▯ ▯▯▯▯▯ ▯▯▯ ▯▯▯▯▯ The fundamental Theorem of Invertible Matrices: version 2▯ Let A be a matrix of size n▯n. The following statements are equivalent. (1) A is invertible. (2) Ax ▯ b has a unique solution for every vector b in R . (3) The equation Ax ▯ ▯ has only the trivial solution. (4) A is row equivalent to the identity matnix I . (5) ▯▯▯▯▯A▯ ▯ n. (6) ▯▯▯▯▯▯▯▯A▯ ▯ ▯. (7) The columns of A are linearly independent. n (8) The columns of A span R . (9) The columns of A form a basis for R . (10) A has n pivot positions. (11) The row vectors of A are linearly independent. (12) The row vectors of A span R . n (13) The row vectors of A form a basis for R . n (14) The equation Ax ▯ b has at least one solution for each b 2 R . (15) There is a square matrix C of size n ▯ n such that CA n I . (16) There is a square matrix D of size n ▯ n such that AD n I . (17) A is invertible. Example. ▯▯▯▯ ▯▯▯▯ ▯▯▯ ▯▯▯▯▯▯▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ; ▯ ; ▯ ▯ ▯ ▯ ▯▯▯▯ ▯ ▯▯▯▯▯ ▯▯▯ R ▯ Solution. ▯▯▯▯▯▯▯▯▯ ▯▯ ▯▯▯ ▯▯▯▯▯▯▯▯▯▯▯ ▯▯▯▯▯▯▯ ▯▯▯ ▯▯▯▯▯▯▯ ▯▯▯▯ ▯ ▯▯▯▯▯ ▯▯ ▯▯▯ ▯▯▯▯▯▯ ▯▯▯▯ ▯▯▯▯▯ ▯▯▯▯▯▯▯ ▯▯ ▯▯▯ ▯▯▯▯▯▯▯ ▯▯▯ ▯▯▯▯ ▯▯ ▯▯ ▯▯▯▯▯▯▯▯▯▯▯▯ ▯ ▯▯▯▯▯ ▯▯▯▯▯▯▯▯▯▯ ▯▯ ▯▯▯▯ ▯ ▯ ▯ ▯ ▯ ▯▯ ▯ ▯ ▯▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯! ▯ ▯ ▯ ▯ ▯: ▯ ▯ ▯ ▯ ▯ ▯▯ ▯▯▯▯ ▯▯ ▯▯▯▯ ▯▯▯▯▯ ▯▯▯▯▯▯▯ ▯▯▯▯▯▯▯ ▯▯▯ ▯▯ ▯▯▯ ▯▯▯▯▯ ▯▯▯ ▯▯▯▯▯▯▯▯▯ ▯▯▯▯▯▯▯ ▯▯ ▯▯▯▯ ▯▯▯▯▯▯ ▯▯▯ ▯▯▯▯▯▯▯▯▯▯▯▯▯ Theorem (the basis theorem). ▯▯▯ H ▯▯ ▯ p▯▯▯▯▯▯▯▯▯▯▯▯ ▯▯▯▯▯▯▯▯ ▯▯ R ▯ ▯▯▯ ▯▯▯▯▯▯▯▯ ▯▯▯▯▯▯▯▯▯▯▯ ▯▯▯ ▯▯ ▯▯▯▯▯▯▯ p ▯▯▯▯▯▯▯▯ ▯▯ H ▯▯ ▯▯▯▯▯▯▯▯▯▯▯▯▯ ▯ ▯▯▯▯▯ ▯▯▯ H▯ ▯▯▯▯ ▯▯▯ ▯▯▯ ▯▯ p ▯▯▯▯▯▯▯▯ ▯▯ H ▯▯▯▯ ▯▯▯▯ H ▯▯ ▯▯▯▯▯▯▯▯▯▯▯▯▯ ▯ ▯▯▯▯▯ ▯▯▯ H▯ Example. ▯▯▯▯▯▯▯ ▯▯▯▯ ▯ ▯▯▯▯▯▯ A ▯▯ ▯▯▯▯ ▯ ▯ ▯ ▯▯▯ ▯▯▯▯▯ ▯▯▯▯▯ ▯▯▯▯▯▯▯▯ ▯▯ ▯▯ ▯▯▯▯ ▯▯▯▯ ▯▯▯A ▯ R ▯ ▯▯▯▯ ▯▯ ▯▯▯ ▯▯▯▯▯▯▯▯▯ ▯▯ ▯▯▯A▯ Solution. ▯▯▯▯▯ A ▯▯▯ ▯▯▯▯▯ ▯▯▯▯▯ ▯▯▯▯▯▯▯▯▯▯ ▯▯ ▯▯▯ ▯▯▯▯▯ ▯▯▯▯▯▯▯▯ ▯▯▯▯▯ ▯▯▯▯▯▯▯ ▯▯▯▯▯▯▯▯ ▯▯▯ a1;a2;3 ▯ ▯▯▯▯▯ ▯▯▯▯ ▯▯▯A▯ ▯▯▯▯▯ ▯▯ ▯▯▯ ▯▯▯▯▯ ▯▯▯▯▯ ▯▯▯▯▯▯▯ ▯▯▯▯▯ ▯▯▯▯▯▯▯ a1;2 ;3 ▯▯▯▯ ▯ ▯▯▯▯▯ ▯▯ R ▯▯▯▯▯▯▯ R 3 3 ▯▯▯ ▯▯▯▯▯▯▯▯▯ ▯▯ ▯▯▯▯ ▯▯▯▯▯▯▯ ▯▯▯▯ ▯▯▯▯▯ ▯▯▯▯▯▯ ▯▯ R ▯▯ ▯ ▯▯▯▯▯▯ ▯▯▯▯ ▯▯▯▯▯▯▯▯ ▯▯ 1 ;2 ;3 ▯ ▯▯▯▯▯ ▯▯▯A ▯ R ▯ ▯▯▯▯▯ ▯▯ ▯▯▯ ▯▯▯▯ ▯▯▯▯▯▯▯ ▯▯ ▯▯▯▯ ▯▯▯▯▯▯A ▯ ▯ ▯ ▯▯▯▯A ▯ ▯ ▯ ▯ ▯ ▯: Coordinate systems n ▯▯▯ H ▯▯ ▯ ▯▯▯▯▯▯▯▯ ▯▯ R ▯ ▯▯▯ ▯▯▯▯ ▯▯▯▯▯▯ ▯▯▯ ▯▯▯▯▯▯▯▯▯ ▯ ▯▯▯▯▯ ▯▯▯ ▯ ▯▯▯▯▯▯▯▯ H▯ ▯▯▯▯▯▯▯ ▯▯ ▯▯▯▯▯▯ ▯ ▯▯▯▯▯▯▯▯ ▯▯▯▯ ▯▯ ▯▯▯▯ ▯▯▯▯ ▯▯▯▯▯▯ ▯▯ H ▯▯▯ ▯▯ ▯▯▯▯▯▯▯ ▯▯ ▯▯▯▯ ▯▯▯ ▯▯▯ ▯▯ ▯ ▯▯▯▯▯▯ ▯▯▯▯▯▯▯▯▯▯▯ ▯▯ ▯▯▯ ▯▯▯▯▯ ▯▯▯▯▯▯▯▯ ▯▯▯▯▯▯▯ ▯▯▯▯▯▯ ▯▯▯▯ B ▯ f1 ;:::pb g ▯▯ ▯ ▯▯▯▯▯ ▯▯▯ H▯ ▯▯▯▯▯▯ ▯▯▯▯ ▯ ▯▯▯▯▯▯ v 2 H ▯▯▯ ▯▯ ▯▯▯▯▯▯▯ ▯▯ ▯▯▯ ▯▯ff▯▯▯▯▯ ▯▯▯▯ v ▯ c b ▯ ▯▯▯ ▯ c b ▯▯▯ v ▯ d b ▯ ▯▯▯ ▯ d b : 1 1 p p 1 1 p p ▯▯▯▯ ▯▯▯▯▯▯▯▯▯▯▯ ▯▯ ▯▯▯▯ ▯ ▯ v ▯ v ▯ ▯1 ▯ 1 ▯1 ▯ ▯▯▯ ▯ pc ▯pd pb : ▯▯▯ b1;:::;p ▯▯▯ ▯▯▯▯▯▯▯▯ ▯▯▯▯▯▯▯▯▯▯▯▯ ▯▯▯▯1 c 1d ▯ ▯;:::pc ▯p ▯ ▯▯ ▯▯▯▯▯▯▯▯ 1 ▯ d1▯ ▯▯▯▯pc ▯pd ▯ De▯nition. Assume that B ▯ fb ;:1:;b gpis a basis for a subspace H. For each vector x 2 H the coordinates of x relative to the basis B are the scalars c ;:::;c such that x ▯ c b ▯ ▯▯▯ ▯ c b . The vector 1 p 1 1 p p ▯ c1 ▯ ▯ c2 ▯
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