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Question 1 The curve y = ax² + bx + c passes through the points (x₁, y₁), (x2, Y2) and (x3, Y3). (i) Show that the coefficients a, b and c form the solution of the system of linear equations with the following augmented matrix: (a) [x²x₁ 1 y₁ 1 y2 x₂x₂ X3 1 y3.

Question 1 The curve y = ax² + bx + c passes through the points (x₁, y₁), (x2, Y2) and (x3, Y3). (i) Show that the coefficients a, b and c form the solution of the system of linear equations with the following augmented matrix: (a) [x²x₁ 1 y₁ 1 y2 x₂x₂ X3 1 y3.

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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Question 1 The curve y = ax² + bx+c passes through the points (x₁, y₁), (x₂, Y₂) and (x3, Y3). (i) Show that the coefficients a, b and c form the solution of the system of linear equations with the following augmented matrix: (a) (b) (ii) Use the result in Question 1(a)(i) and Gauss-Jordan elimination method to solve the values of a, b and c for which the curve y = ax² + bx + c passes through the points (0,4), (2,10) and (3,19). [1 2 0 21 Discuss the existence and uniqueness of solutions to the linear systems with the augmented matrices shown below: A 0 2 1 Lo 00 [X₁ X₁ x²x₂ [x3 x3 1 1. 1 y₁ 1 y2 1 уз. B = 0 100 220 0 2 1 1 0 0J 100 C = 0 20 21 2 1 1 0 1 0
Transcribed Image Text:Question 1 The curve y = ax² + bx+c passes through the points (x₁, y₁), (x₂, Y₂) and (x3, Y3). (i) Show that the coefficients a, b and c form the solution of the system of linear equations with the following augmented matrix: (a) (b) (ii) Use the result in Question 1(a)(i) and Gauss-Jordan elimination method to solve the values of a, b and c for which the curve y = ax² + bx + c passes through the points (0,4), (2,10) and (3,19). [1 2 0 21 Discuss the existence and uniqueness of solutions to the linear systems with the augmented matrices shown below: A 0 2 1 Lo 00 [X₁ X₁ x²x₂ [x3 x3 1 1. 1 y₁ 1 y2 1 уз. B = 0 100 220 0 2 1 1 0 0J 100 C = 0 20 21 2 1 1 0 1 0
(c) If S = {V₁, V₂, ..., Vn} is a set of vectors in a finite-dimensional vector space V, then S is called a basis for V if S is linearly independent and every vector b = (b₁,b₂, ..., bn) in V can be expressed as b = c₁v₁ + C₂V₂ + ... + CnVn where C₁, C₂, ..., Cn are scalars. Calculate the basis for the solution space of the following system of linear equations and verify your answer. x₁ + 2x3 x4 = 0 -x₂ + 2x4 = 0
Transcribed Image Text:(c) If S = {V₁, V₂, ..., Vn} is a set of vectors in a finite-dimensional vector space V, then S is called a basis for V if S is linearly independent and every vector b = (b₁,b₂, ..., bn) in V can be expressed as b = c₁v₁ + C₂V₂ + ... + CnVn where C₁, C₂, ..., Cn are scalars. Calculate the basis for the solution space of the following system of linear equations and verify your answer. x₁ + 2x3 x4 = 0 -x₂ + 2x4 = 0
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