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)