I have the following problem for my algebra homework: Given two systems of linear equations with coefficients in R is it possible to find a system of linear equations that describes the a) sum b) intersection c) difference of the sets of solutions of those systems. Prove or disprove.
My way of thinking is as follows: For simplification I assume the solutions are in R2 (but i guess this can be extended to any number of dimensions). So a solution of a system of equations can be either a line or a point. If solutions to both are points then:
a) is not possible because it is not possible to find a SOE that resolves to two points. b) is possible, if the solutions are not the same then the intersection is an empty set, otherwise it is a point. c) is also possible as it is either a point or an empty set
If a solution to one of them is a point and to the other a line:
a) not possible, would only exist if the point was on the line b) possible, either an empty set or a point c) not sure how to interpret it
If solutions to both systems are lines: a) not possible, would only exist if it was the same line b) possible, it can be an empty set, a point or a line c) not sure again
To be honest I am not even sure if what I am saying makes any sense ;) Any help will be greatly appreciated :)