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MATH125 (103)
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Lecture 21

MATH125 Lecture 21: 21

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Department
Mathematics
Course
MATH125
Professor
Nikita
Semester
Winter

Description
▯▯▯▯▯▯▯ ▯▯▯ ▯▯▯▯▯ ▯▯ ▯▯▯▯▯ Characterization of invertible matrices ▯▯▯▯ ▯▯▯▯▯▯▯ ▯▯▯▯▯▯▯▯ ▯ ▯▯▯▯▯▯ ▯▯ ▯▯ff▯▯▯▯▯ ▯▯▯▯▯▯▯▯ ▯▯▯▯ ▯ ▯▯▯▯▯▯ ▯▯▯▯▯▯ A ▯▯ ▯▯▯▯▯▯▯▯▯▯▯ The invertible matrix theorem. Let A be a square matrix of size n ▯ n. Then the following statements are equivalent. (1) A is invertible. (2) A is row equivalent to the identity matrix In. (3) The equation Ax ▯ ▯ has only the trivial solution. (4) The columns of A are linearly independent. n (5) The equation Ax ▯ b has at least one solution for each b 2 R . (6) The columns of A span R . n (7) There is a square matrix C of size n ▯ n such that CA ▯ I . n (8) There is a square matrix D of size n ▯ n such that AD ▯ I . n (9) A is invertible. ▯▯▯ ▯▯▯▯▯▯▯▯▯▯ ▯▯▯▯▯▯ ▯▯▯▯▯▯▯ ▯▯▯▯▯▯▯ ▯▯▯ ▯▯▯ ▯▯ ▯▯▯ n ▯ n ▯▯▯▯▯▯▯▯ ▯▯▯▯ ▯▯▯ ▯▯▯▯▯▯▯▯ ▯▯▯▯▯▯▯▯ ▯▯▯ ▯▯▯▯▯▯▯▯▯▯ ▯▯▯ ▯▯▯▯▯▯▯▯▯▯▯▯ ▯▯▯▯▯▯▯▯ ▯▯▯ ▯▯▯ ▯▯▯ ▯▯▯▯▯▯▯▯▯▯ ▯▯▯ ▯▯▯▯▯▯▯▯▯ ▯▯▯▯▯▯▯▯▯ x▯▯▯▯ ▯▯▯▯▯▯▯▯▯▯ ▯▯▯▯▯▯ ▯▯▯▯▯▯▯▯▯ ▯▯▯ ▯▯▯▯ ▯▯▯▯ ▯▯▯▯▯▯▯ ▯▯▯▯▯▯▯ ▯▯ ▯▯▯▯▯▯▯ ▯▯▯▯ ▯▯▯▯▯▯▯▯▯ ▯▯▯▯▯ ▯▯ ▯▯▯ ▯▯▯▯▯▯ ▯▯▯▯▯▯▯ ▯▯▯ ▯▯▯▯▯▯▯ ▯▯ ▯ ▯▯▯▯▯▯▯▯▯ ▯ ▯▯▯▯▯▯ ▯▯▯▯ ▯▯▯ ▯▯▯▯▯▯▯▯▯▯ ▯▯▯ ▯▯▯ ▯▯▯▯ ▯▯ ▯ ▯▯▯▯▯▯▯▯ ▯▯ ▯▯▯▯▯▯ ▯▯ ▯▯▯▯▯▯▯▯▯ ▯▯▯▯▯▯▯▯▯▯▯▯▯▯ ▯▯ ▯▯▯ ▯▯▯▯▯▯▯▯▯ ▯▯▯▯▯▯▯▯ ▯▯ ▯▯▯▯▯ ▯▯▯ ▯▯▯▯▯▯ ▯▯▯▯▯▯▯ ▯▯▯ ▯▯▯▯▯▯▯ ▯▯▯ ▯▯▯▯▯▯▯▯▯▯▯ ▯▯▯▯ ▯▯▯▯ ▯▯ ▯ ▯▯▯▯▯ ▯▯▯ ▯ ▯▯▯▯▯▯▯▯ ▯▯ ▯▯▯▯ ▯▯▯▯▯▯▯ ▯▯▯▯ ▯▯▯ ▯▯▯▯ ▯▯ ▯▯▯▯▯▯▯▯▯ ▯▯▯▯▯▯▯ ▯▯▯ ▯▯▯▯ ▯▯▯▯▯ ▯▯▯ ▯▯▯▯▯▯▯ n n De▯nition. ▯ ▯▯▯▯▯▯▯▯ ▯▯ R ▯▯ ▯▯▯ ▯▯▯▯▯▯ H ▯ R ▯▯▯▯ ▯▯▯ ▯▯▯ ▯▯▯▯▯▯▯▯▯ ▯▯▯▯▯▯▯▯▯▯▯ ▯▯▯ ▯▯▯ ▯▯▯▯ ▯▯▯▯▯▯ ▯▯ ▯▯ H▯ ▯▯▯ ▯▯▯ ▯▯▯▯ u;v 2 H ▯▯▯▯▯ ▯▯▯ u ▯ v ▯▯ ▯▯▯▯ ▯▯ H▯ ▯▯▯ ▯▯▯ ▯▯▯▯ u 2 H ▯▯▯ ▯▯▯▯ ▯▯▯▯▯▯ c ▯▯▯ ▯▯▯▯▯▯ cu ▯▯ ▯▯▯▯ ▯▯ H▯ Remark. In words, we say that H is closed under addition and scalar multiplication and contains zero. n n Example. ▯▯▯ u;v 2 R ▯ ▯▯▯▯ ▯▯▯▯fu;v g ▯▯ ▯ ▯▯▯▯▯▯▯▯ ▯▯ R ▯ ▯▯▯ ▯▯▯▯▯ ▯▯ ▯▯▯▯ ▯▯ ▯▯▯▯▯▯ ▯▯▯▯▯ ▯▯▯▯▯▯▯▯▯▯▯ ▯▯ ▯▯▯▯ ▯▯▯▯▯▯ ▯▯▯▯ ▯▯▯f u;vg ▯▯▯▯▯▯▯▯ ▯▯ ▯▯▯ ▯▯▯▯▯▯ ▯▯▯▯▯▯▯▯▯▯▯▯ au ▯ bv ▯▯▯▯▯ a;b ▯▯▯ ▯▯▯▯▯▯▯▯ ▯▯▯ ▯▯ ▯▯▯▯ ▯u ▯ ▯v ▯ ▯ ▯▯ ▯▯ ▯▯▯▯fu;v g▯ ▯▯▯ ▯▯▯▯ ▯▯▯ ▯▯▯▯▯▯▯ ▯▯ ▯▯▯▯▯ x ▯ au ▯ bv ▯▯▯ y ▯ cu ▯ dv▯ ▯▯▯▯ x ▯ y ▯ ▯au ▯ bv▯ ▯ ▯cu ▯ dv▯ ▯ ▯a ▯ c▯u ▯ ▯b ▯ d▯v; ▯▯▯▯▯ x ▯ y 2 ▯▯▯▯fu;v g▯ ▯▯▯ ▯▯▯ x ▯ au▯bv 2 ▯▯▯▯fu;v g ▯▯▯ ▯▯▯ c ▯▯ ▯ ▯▯▯▯▯▯▯ ▯▯▯▯ ▯▯ ▯▯▯▯ cx ▯ c▯au ▯ bv▯ ▯ ▯ca▯u ▯ ▯cb▯v 2 ▯▯▯▯fu;v g: Example. ▯ ▯▯▯▯ ▯▯ R ▯▯▯ ▯▯▯▯▯▯▯ ▯▯▯▯▯▯▯ ▯▯▯ ▯▯▯▯▯▯ ▯▯ ▯▯▯ ▯ ▯▯▯▯▯▯▯▯ ▯▯▯▯▯▯▯ ▯▯ ▯▯▯▯ ▯▯▯ ▯▯▯▯▯▯▯ ▯▯▯ ▯▯▯▯ ▯▯▯▯▯▯▯ 3 Example. ▯ ▯▯▯▯▯ ▯▯ R ▯▯▯▯▯▯▯ ▯▯▯▯▯▯▯ ▯▯▯ ▯▯▯▯▯▯ ▯▯ ▯ ▯▯▯▯▯▯ ▯▯▯▯ ▯▯▯▯▯▯ ▯▯▯▯▯▯▯ ▯▯1 d2; d ▯▯ ▯▯▯▯▯▯▯▯▯ ▯▯▯▯▯▯▯ ▯▯ ▯▯▯▯ ▯▯▯▯▯▯ ▯▯▯ ▯▯▯▯▯▯ ▯▯▯▯▯▯▯▯ ▯▯ ▯▯▯ ▯▯▯▯▯ ▯▯
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