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Lecture 22

MATH125 Lecture 22: 22

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Department
Mathematics
Course
MATH125
Professor
Nikita
Semester
Winter

Description
▯▯▯▯▯▯▯ ▯▯▯ ▯▯▯▯▯ ▯▯ ▯▯▯▯▯ Subspaces associated with matrices ▯▯ ▯▯▯▯▯▯▯▯ ▯▯▯▯ ▯▯ ▯▯▯▯▯▯▯▯▯ ▯▯▯▯▯ ▯▯ ▯▯▯ ▯▯▯▯▯▯▯▯▯ ▯▯▯▯▯▯▯▯▯ ▯▯▯▯ ▯▯▯▯▯▯▯▯▯▯ ▯▯ ▯▯▯▯▯▯▯▯▯ De▯nition. Let A be a matrix of size m ▯ n. The column subspace of a matrix A is the set of all linear combinations of the columns of A. ▯▯▯▯ ▯▯▯ ▯▯▯▯▯▯▯▯ ▯▯ ▯▯▯▯ ▯▯▯▯ ▯▯ ▯▯ ▯ ▯▯▯▯▯▯ ▯▯▯▯▯▯▯▯ ▯▯▯▯ ▯▯ ▯▯ ▯▯▯▯▯▯▯ ▯▯ ▯▯▯A▯ ▯▯ ▯▯▯▯▯▯▯ ▯▯▯▯ ▯▯▯▯▯▯▯▯▯ ▯▯▯▯ ▯▯ A ▯ ▯a a ▯▯▯ a ▯ ▯▯▯▯▯ a ;:::;a 1 2 n 1 n ▯▯▯ ▯▯▯▯▯▯▯ ▯▯ A ▯▯▯▯ ▯▯▯A ▯▯ ▯▯▯ ▯▯▯▯ ▯▯ ▯▯▯1fa ;:n:;a g▯ Example. ▯▯▯ ▯ ▯ ▯ ▯▯ ▯▯ A ▯ ▯ ▯▯ ▯ ▯▯ ▯ ▯▯ ▯ ▯ ▯▯▯ ▯▯▯ ▯ ▯ ▯ b ▯▯ ▯ ▯ : ▯▯ ▯▯▯▯▯▯▯▯▯ ▯▯▯▯▯▯▯ b ▯▯ ▯▯ ▯▯▯A▯ Solution. ▯▯▯ ▯▯▯▯▯▯ b ▯▯ ▯▯ ▯▯▯A ▯▯ ▯▯▯ ▯▯▯▯ ▯▯ ▯▯ ▯▯▯ ▯▯ ▯▯▯▯▯▯▯ ▯▯ ▯ ▯▯▯▯▯▯ ▯▯▯▯▯▯▯▯▯▯▯ ▯▯ ▯▯▯▯▯▯▯ ▯▯ A▯ ▯▯▯▯ ▯▯ ▯▯▯ ▯▯▯▯ ▯▯ ▯▯▯ ▯▯▯▯▯▯▯▯ Ax ▯ b ▯▯▯ ▯ ▯▯▯▯▯▯▯▯▯ ▯▯▯ ▯▯▯▯▯▯▯▯▯ ▯▯▯▯▯▯ ▯▯ ▯▯▯▯ ▯▯▯▯▯▯▯▯ ▯▯ ▯ ▯ ▯ ▯▯ ▯▯ j ▯ ▯ ▯▯ ▯ ▯▯ j ▯ ▯: ▯▯ ▯ ▯ j ▯▯ ▯▯ ▯▯▯▯▯▯▯ ▯▯▯▯ ▯▯ ▯ ▯ ▯ ▯▯ ▯▯ j ▯ ▯ ▯ ▯ ▯▯ ▯▯▯ j ▯▯ : ▯ ▯ ▯ j ▯ ▯▯ ▯▯▯▯▯▯▯ ▯▯▯▯ ▯▯▯ ▯▯▯▯▯▯▯▯ ▯▯▯ ▯ ▯▯▯▯▯▯▯▯ ▯▯▯ ▯▯▯▯ ▯▯ ▯▯▯ ▯▯▯▯▯▯▯▯▯ ▯▯▯▯ ▯▯▯▯▯▯▯▯▯ ▯▯▯▯▯▯▯ ▯▯ ▯▯▯▯ ▯ ▯▯▯▯ ▯▯▯▯▯▯▯▯▯▯ ▯▯▯▯▯ b 2 ▯▯▯A▯ De▯nition. Let A be a matrix of size m ▯ n. The null subspace of a matrix A is the set of all solutions of the equation Ax ▯ ▯. It is denoted by ▯▯▯A. ▯▯ ▯▯▯▯▯▯▯ ▯▯ ▯▯▯▯ ▯▯ ▯▯▯▯▯▯ ▯▯▯▯ ▯▯▯ ▯▯▯ ▯▯ ▯▯▯ ▯▯▯▯▯▯▯▯▯ ▯▯ ▯▯▯ ▯▯▯▯▯ ▯▯▯▯ Ax ▯ ▯ ▯▯▯▯▯▯▯▯ ▯▯▯▯▯ ▯▯▯▯▯▯▯▯▯▯ ▯▯ ▯▯▯ ▯▯▯▯▯▯▯▯▯ ▯▯ ▯ ▯▯▯▯▯▯▯▯ ▯▯ ▯▯▯▯▯ ▯▯ ▯▯▯▯▯ ▯▯▯▯ ▯▯▯A ▯▯ ▯▯▯▯▯▯ ▯ ▯▯▯▯▯▯▯▯ ▯▯ R ▯ ▯▯▯ ▯▯ ▯▯▯▯▯ ▯▯▯▯▯ ▯▯▯▯▯▯▯▯▯▯▯ ▯▯▯ ▯ 2 ▯▯▯A ▯▯▯▯▯▯▯ A ▯ ▯ ▯ ▯▯ ▯▯▯ ▯▯▯ u;v 2 ▯▯▯A▯ ▯▯▯▯▯ ▯▯ ▯▯▯▯▯▯▯▯▯ Au ▯ Av ▯ ▯▯ ▯▯▯▯ ▯▯▯ ▯▯▯ u ▯ v ▯▯▯ ▯▯▯ ▯▯ ▯▯▯▯▯▯▯ A▯u ▯ v▯▯ ▯▯ ▯▯▯▯ A▯u ▯ v▯ ▯ Au ▯ Av ▯ ▯ ▯ ▯ ▯ ▯: ▯▯▯ ▯▯▯▯▯▯▯▯▯▯ ▯▯ u 2 ▯▯▯A ▯▯▯ c ▯▯ ▯ ▯▯▯▯▯▯ ▯▯ ▯▯ ▯▯▯▯ ▯▯ ▯▯▯ ▯▯▯▯ cu 2 ▯▯▯A▯ Remark. ▯▯ ▯▯▯▯ ▯▯▯▯▯▯▯ ▯ ▯▯▯▯▯▯ u ▯▯ ▯▯ ▯▯▯A ▯▯▯ ▯▯▯▯▯ ▯▯ ▯▯▯▯▯▯▯ Au▯ ▯▯ Au ▯▯ ▯▯▯▯▯ ▯▯▯▯ u 2 ▯▯▯A▯ ▯▯▯▯▯▯▯▯▯ u ▯▯ ▯▯▯ ▯▯ ▯▯▯A▯ De▯nition. Let A be a matrix of size m ▯ n. The row space of A is the subspace in R spanned by the rows of A. Notation: ▯▯▯▯A▯. Example. ▯▯▯▯▯▯▯▯ ▯▯▯ ▯▯▯▯▯▯ ▯ ▯ ▯ ▯▯ A ▯ ▯ ▯ ▯ ▯ : ▯ ▯▯ ▯▯▯▯▯▯▯▯▯ ▯▯ [ ] ▯ w ▯ ▯ ▯▯ ▯▯ ▯▯▯ ▯▯▯ ▯▯▯▯▯ ▯▯ A▯ Solution. ▯▯▯ ▯▯▯ ▯▯▯▯▯ ▯▯ ▯▯▯▯▯▯▯ ▯▯ ▯▯▯▯▯ ▯▯▯▯▯▯▯ [ ] [ ] [ ] v1▯ ▯ ; v2▯ ▯ ; v3▯ ▯ : ▯▯ ▯ ▯▯ ▯▯▯▯▯ w ▯▯ ▯▯ ▯▯▯ ▯▯▯ ▯▯▯▯▯ ▯▯ ▯▯▯ ▯▯▯▯ ▯▯ ▯▯▯ ▯▯▯▯▯▯ ▯▯▯▯▯▯ ▯▯▯▯▯ ▯▯▯▯▯▯▯▯▯ ▯▯▯▯▯▯ ▯▯ [ ] ▯ ▯ ▯ j ▯ ▯▯ ▯ ▯▯ j ▯ ▯▯▯ ▯ ▯▯▯▯▯▯▯▯▯ ▯▯▯▯▯▯▯ ▯▯ ▯▯▯ ▯▯▯▯▯▯▯ ▯▯▯▯▯▯▯ ▯▯▯▯ ▯▯ ▯▯▯ ▯▯▯▯▯▯ ▯▯▯ ▯▯▯▯ ▯▯▯ ▯▯▯▯▯▯ ▯▯▯ ▯ ▯▯▯▯▯▯▯▯▯ Remark. ▯▯▯▯▯▯▯ ▯▯▯▯▯▯▯▯▯ ▯▯ ▯▯ ▯▯▯▯ ▯▯ ▯▯▯ ▯▯▯▯ ▯▯▯▯▯ ▯▯▯▯▯▯ [ ] a b ▯▯ R ▯▯ ▯▯ ▯▯▯ ▯▯▯ ▯▯▯▯▯ ▯▯ A▯ ▯▯▯▯▯ ▯▯ ▯▯▯▯ ▯▯▯▯▯▯▯ ▯▯▯▯A▯ ▯ R ▯ T Example. ▯▯▯▯▯▯▯▯ ▯▯▯▯A▯ ▯▯▯▯▯▯▯▯▯ ▯▯▯▯ ▯▯▯ ▯▯▯▯▯▯ ▯▯▯▯▯ ▯▯ A ▯ Theorem. Let B be any matrix that is row equivalent to a matrix A. Then ▯▯▯▯B▯ ▯ ▯▯▯▯A▯. Proof. ▯▯▯ ▯▯▯▯▯▯ A ▯▯▯ ▯▯ ▯▯▯▯▯▯▯▯▯▯▯ ▯▯▯▯ B ▯▯ ▯ ▯▯▯▯▯▯▯▯ ▯▯ ▯▯▯ ▯▯▯▯▯▯▯▯▯▯▯ ▯▯▯▯▯▯▯▯▯▯▯▯▯ ▯▯▯ ▯▯▯▯ ▯▯ B ▯▯▯ ▯▯▯▯▯▯ ▯▯▯▯▯▯▯▯▯▯▯▯ ▯▯ ▯▯▯ ▯▯▯▯ ▯▯ A▯ ▯▯▯ ▯▯▯▯ ▯▯▯▯▯▯▯▯ ▯▯▯▯▯ ▯▯▯▯ ▯▯▯ ▯▯▯▯ ▯▯ A ▯▯▯ ▯▯▯▯▯▯ ▯▯▯▯▯▯▯▯▯▯▯▯ ▯▯ ▯▯▯ ▯▯▯ ▯▯ B▯ ▯▯▯▯ ▯▯▯ ▯▯▯ ▯▯ ▯▯▯ ▯▯▯▯▯▯ ▯▯▯▯▯▯▯▯▯▯▯▯ ▯▯ ▯▯▯▯ ▯▯ A ▯▯▯▯▯▯▯▯▯ ▯▯▯▯ ▯▯▯ ▯▯▯ ▯▯ ▯▯▯ ▯▯▯▯▯▯ ▯▯▯▯▯▯▯▯▯▯▯▯ ▯▯ ▯▯▯ ▯▯▯▯ ▯▯ B ▯▯▯ ▯▯ ▯▯▯ ▯▯▯▯▯ ▯ Basis for a subspace n ▯ ▯▯▯▯▯▯▯▯ ▯▯ R ▯▯▯▯▯▯▯▯▯ ▯▯▯▯▯▯▯▯ ▯▯ ▯▯▯▯▯▯▯ ▯▯▯▯▯▯ ▯▯ ▯▯▯▯▯▯▯▯ ▯▯▯▯▯▯▯ ▯▯▯▯ ▯▯ ▯▯▯ ▯▯▯▯▯▯▯▯ ▯▯▯▯▯▯▯▯▯ ▯ ▯▯▯▯▯▯▯▯ ▯▯▯ ▯▯▯▯▯▯▯ ▯▯ ▯▯▯▯▯▯▯ ▯▯▯▯ ▯ ▯▯▯▯▯ ▯▯▯▯▯▯ ▯▯ ▯▯▯▯▯▯▯ ▯▯▯▯ ▯▯▯▯ ▯▯▯ ▯▯▯▯▯▯▯▯▯ ▯▯▯ ▯▯▯▯▯▯▯ ▯▯▯ ▯▯▯ ▯▯▯▯▯▯▯ n De▯nition. ▯ ▯▯▯▯▯ ▯▯▯ ▯ ▯▯▯▯▯▯▯▯ H ▯▯ R ▯▯ ▯ ▯▯▯▯▯▯▯▯ ▯▯▯▯▯▯▯▯▯▯▯ ▯▯▯ ▯▯ ▯▯▯▯▯▯▯ ▯▯ H ▯▯▯▯ ▯▯▯▯ H▯ ▯▯▯▯▯ ▯ ▯▯▯▯▯▯ fv1;:::;vpg ▯ H ▯▯ ▯ ▯▯▯▯▯ ▯▯ ▯▯▯ ▯▯▯▯▯ ▯▯▯▯▯▯▯ ▯▯▯ ▯▯▯▯▯▯▯▯ ▯▯▯▯▯▯▯▯▯▯▯▯ ▯▯▯ ▯▯▯▯ ▯▯▯▯ H▯ ▯▯▯▯ ▯▯▯▯▯ ▯▯▯▯▯▯ ▯▯ H ▯▯▯ ▯▯ ▯▯▯▯▯▯▯ ▯▯ ▯ ▯▯▯▯▯▯ ▯▯▯▯▯▯▯▯▯▯▯ ▯▯ ▯▯▯ ▯▯▯▯▯▯▯ 1 ;:::pv ▯ Example. ▯▯▯ A ▯▯ ▯▯ ▯▯▯▯▯▯▯▯▯▯ ▯▯▯▯▯▯ ▯▯ ▯▯▯▯ n ▯nn▯ ▯▯▯
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