question 1 a the curve y ax bx c passes through the points x y x2 y an

Question

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)