Vector Space Solutions for Systems: Explained Here

[EvLer]

Hi everyone,
general question: is a solution set for a particular system a vector space? I know it can be if there is a unique solution, but is it generally true?
Could someone explain, please?

Thanks.

 

[AKG]

No. Vector spaces are closed under scalar multiplication. If b is a scalar not equal to 1, Y is non-zero, and X is a solution of AX = Y, then:

A(bX) = b(AX) = bY is not equal to Y, so (bX) is not a solution, so the set of solutions is not closed under scalar multiplication, so the set of solutions is not a vector space. Perhaps I've misinterpreted your question. If there is a unique solution, then there would only be that 1 element of the vector space. The only vector space that has only one element is the degenerate vector space {0}.

 

[HallsofIvy]

For a particular system? Do you mean a system of linear equations?

The solution set of a system of homogenous equations is a subspace.

If the system consists of n independent equations in n unknowns, then it is just the 0 vector but if the rank is lower than the number of unknowns, then it is a non-trivial subspace of R^n[/sub].