Question

6) (after 2.3) Consider the infinite system of linear equations in two variables given by ax + by = 0 where (a, b) moves along the unit circle in the plane. (a) How many solutions does this system have? (b) What is the smallest number of equations in the above system that have the same solution set? Write down two separate such linear systems, in vector form. (c) What happens to the infinite linear system if you add the equation 0x + 0y = 0 to it? (d) What happens to the infinite linear system if by accident one of the equations was recorded as ax + by = 0.00001? Explain all your answers in words.

Answer 1

The given system of linear equations has infinitely many solutions as (a, b) moves along the unit circle in the plane. The smallest number of equations in the system that have the same solution set is 1. If we add the equation 0x + 0y = 0 to the system, it doesn't change the solution set and if one of the equations was recorded as ax + by = 0.00001, it would create a slight deviation from the unit circle.

Step-by-step explanation:

The given system of linear equations is ax + by = 0, where (a, b) moves along the unit circle in the plane. Let's analyze each part of the question:

(a) The system of equations has infinitely many solutions. This is because for every point on the unit circle, we can find a unique solution that satisfies the equation.

(b) The smallest number of equations in the system that have the same solution set is 1. This means that a single equation is enough to represent the solution set of the entire system. Two such linear systems in vector form could be:

System 1: [x, y] = [1, 0]

System 2: [x, y] = [-1, 0]

(c) If we add the equation 0x + 0y = 0 to the system, it doesn't change the solution set. This equation is a trivial one, as it doesn't provide any additional information.

(d) If one of the equations was recorded as ax + by = 0.00001, it would create a slight deviation from the unit circle. The solution set would still be infinite, but the points would no longer lie exactly on the unit circle. This is because the equation no longer represents the unit circle.

Answer 2

Step-by-step explanation:

Given infinite system of linear equations is ax + by = 0

when (a,b) moves along unit circle in plane.

a) system having unique system (0, 0)

Since two of equation in thus system will be

`1.x+0.y=0\\x=0`

and

`0.x+1.y=0\\y=0`

It is clear that x = 0, y= 0 is the only solution

b) Linear independent solution in this system gives some set of solutions

`1.x+0.y=0\\\x=0`

and

`0.x+1.y=0\\y=0`

Vector form is

`\left[\begin{array}{ccc}1&0\\0&1\end{array}\right] =I`

c) for this equation if add 0x +0y = 0 to system , Nothing will change

Because [0,0] satisfies that equation

d) If one of the equation is ax + by = 0.00001

where 0.00001 is small positive number

so, the system will be inconsistent

Therefore, the system will have no solution.