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O Show that the following set of matrices is closed under linear combina- tions. a + 2b a+b a b -{(a + 2b a-3 "5") = a,bER}. : 3b 0 S₁ = i.e. Show that c₁ M₁ + c₂M₂ E S₁ for all choices M₁, M₂ E S₁ and all scalars C₁, C₂ € R.

O Show that the following set of matrices is closed under linear combina- tions. a + 2b a+b a b -{(a + 2b a-3 "5") = a,bER}. : 3b 0 S₁ = i.e. Show that c₁ M₁ + c₂M₂ E S₁ for all choices M₁, M₂ E S₁ and all scalars C₁, C₂ € R.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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(a) Show that the following set of matrices is closed under linear combina- tions. a=b): a,bER}. i.e. Show that c₁M₁ + c₂M₂ € S₁ for all choices M₁, M₂ € S₁ and all scalars C₁, C₂ € R. S₁ is {(ª + ² a + 2b a+b a - 3b (b) Show that the following set of matrices is closed under affine combina- tions. S3 = a -{(82): QbER}. a, S₂ == i.e. Show that c₁ M₁+c₂M₂ € S₂ for all choices M₁, M₂ € S₂ and C₁, C₂ € R satisfying c₁ + C₂ = 1. (c) Show that the following set of matrices is not closed under affine combi- nations. {(66%) -2²=0}. : x,y ER, y - x²
Transcribed Image Text:(a) Show that the following set of matrices is closed under linear combina- tions. a=b): a,bER}. i.e. Show that c₁M₁ + c₂M₂ € S₁ for all choices M₁, M₂ € S₁ and all scalars C₁, C₂ € R. S₁ is {(ª + ² a + 2b a+b a - 3b (b) Show that the following set of matrices is closed under affine combina- tions. S3 = a -{(82): QbER}. a, S₂ == i.e. Show that c₁ M₁+c₂M₂ € S₂ for all choices M₁, M₂ € S₂ and C₁, C₂ € R satisfying c₁ + C₂ = 1. (c) Show that the following set of matrices is not closed under affine combi- nations. {(66%) -2²=0}. : x,y ER, y - x²
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