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.

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

and

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

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

and

Vector form is

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.