Question 1 (a) 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: (b) (2 marks) 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). (6 marks) Discuss the existence and uniqueness of solutions to the linear systems with the augmented matrices shown below: [1 2 0 2 2 1 1 o o 1 [x²x₁ 1 ₁] x²x₂ 1 Y₂ [x3 x3 1 уз A = 0 Lo [1 2 0 21 B 0 2 1 1 C Lo o o ol [1 2 0 21 1 1 0 1 0 2 Lo 0J (6 marks)/n(c) If S = {V₁, V₂2, ..., Vn} is a set of vectors in a finite-dimensional vector space V, then S is called a basis for Vif S is linearly independent and every vector b = (b₁,b₂, ..., bn) in V can be expressed as b = c₁v₁ + C₂V₂ + ... + C₂ V₂ 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 (11 marks)

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