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4. Let (*) be the following homogeneous system of (n-1) linear equations in n unknows F: All X₁ + A₁₂ X₂ + + An Xn = 0. A₂1 X₁ + A₂₂X₂ + ... + Cl₂n Xn = 0. a22X2 7 An-1,1X₁+ an-1,2 X₂ + + Olm-1.n Xn = 0. over it! (*) Is Suppose that no equation in this system linear ambination of the other equations. For ea =1,2,.., n, let Ai & Max) (F) be the matrix obtained by deleting the ith column of t coefficient matrix of (*), and let (i = (-1) det (Ai).

4. Let (*) be the following homogeneous system of (n-1) linear equations in n unknows F: All X₁ + A₁₂ X₂ + + An Xn = 0. A₂1 X₁ + A₂₂X₂ + ... + Cl₂n Xn = 0. a22X2 7 An-1,1X₁+ an-1,2 X₂ + + Olm-1.n Xn = 0. over it! (*) Is Suppose that no equation in this system linear ambination of the other equations. For ea =1,2,.., n, let Ai & Max) (F) be the matrix obtained by deleting the ith column of t coefficient matrix of (*), and let (i = (-1) det (Ai).

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter2: Systems Of Linear Equations
Section2.5: Iterative Methods For Solving Linear Systems
Problem 23EQ
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4. Let (*) be the following homogeneous system F: of (n-1) linear equations in n unknows All X₁ + A₁₂ X₂ t + An Xn = 0. a22X2 A21X₁ + a₂₂X₂ + ... + Cl₂n Xn = 0. An-1,1X₁+ an-1, 2X₂ + + On-in Xn = 0. ,2 Is Suppose that no equation in this system linear combination of the other equations. For each =1,2,.nlet Ai & Ma-x) (F) be the matrix obtained by deleting the ith column of the coefficient matrix of (*), and let det (Ai). Ci over it! = (*) (-1) Prove the following statements (a) Not all of C₁,..., Cn are 0. (5) The solution space of the system (*) is olle-dimen (C) A vector (X₁. --, Xn) € F" is a -sional, solution to the system (*) iff (X₁...., Xn) = t (C₁, , (n) a ***/ for some te F.
Transcribed Image Text:4. Let (*) be the following homogeneous system F: of (n-1) linear equations in n unknows All X₁ + A₁₂ X₂ t + An Xn = 0. a22X2 A21X₁ + a₂₂X₂ + ... + Cl₂n Xn = 0. An-1,1X₁+ an-1, 2X₂ + + On-in Xn = 0. ,2 Is Suppose that no equation in this system linear combination of the other equations. For each =1,2,.nlet Ai & Ma-x) (F) be the matrix obtained by deleting the ith column of the coefficient matrix of (*), and let det (Ai). Ci over it! = (*) (-1) Prove the following statements (a) Not all of C₁,..., Cn are 0. (5) The solution space of the system (*) is olle-dimen (C) A vector (X₁. --, Xn) € F" is a -sional, solution to the system (*) iff (X₁...., Xn) = t (C₁, , (n) a ***/ for some te F.
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