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In the next two questions, you will explore two important subspaces associated to a system of linear equations. We will revisit these two subspaces later on in the course. Presently, the notation will be quite cumbersome and difficult. When we develop more theory, the following two questions will have very simple proofs. Suppose that v = (1, 2,...,xn) and v' = (xx,...,xn) are solutions of the system: a11x1 + 12x2 + ... + ann a21x1 + a222 + + a2n Xn aklı ak22 + + akn Xn Prove that the following are also solutions of the system. 1. Av for A E R 2. v v' Conclude that the set of solutions is a subspace of R" = 0 0 0

In the next two questions, you will explore two important subspaces associated to a system of linear equations. We will revisit these two subspaces later on in the course. Presently, the notation will be quite cumbersome and difficult. When we develop more theory, the following two questions will have very simple proofs. Suppose that v = (1, 2,...,xn) and v' = (xx,...,xn) are solutions of the system: a11x1 + 12x2 + ... + ann a21x1 + a222 + + a2n Xn aklı ak22 + + akn Xn Prove that the following are also solutions of the system. 1. Av for A E R 2. v v' Conclude that the set of solutions is a subspace of R" = 0 0 0

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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In the next two questions, you will explore two important subspaces associated to a system of linear equations. We will revisit these two subspaces later on in the course. Presently, the notation will be quite cumbersome and difficult. When we develop more theory, the following two questions will have very simple proofs. Suppose that v = (1, 2,...,xn) and v' = (xx,...,xn) are solutions of the system: a111a1222 + + ain n = 0 + a2n In a21x1 + a22₂ + 0 ak11 ak22 + + akn In Prove that the following are also solutions of the system. 1. Av for A E R 2. - v' Conclude that the set of solutions is a subspace of R". 0
Transcribed Image Text:In the next two questions, you will explore two important subspaces associated to a system of linear equations. We will revisit these two subspaces later on in the course. Presently, the notation will be quite cumbersome and difficult. When we develop more theory, the following two questions will have very simple proofs. Suppose that v = (1, 2,...,xn) and v' = (xx,...,xn) are solutions of the system: a111a1222 + + ain n = 0 + a2n In a21x1 + a22₂ + 0 ak11 ak22 + + akn In Prove that the following are also solutions of the system. 1. Av for A E R 2. - v' Conclude that the set of solutions is a subspace of R". 0
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