# Thread: Prove the solution set of a system of homogeneous equations is a vector space

1. ## Prove the solution set of a system of homogeneous equations is a vector space

"Show that the solution set of a homogeneous system of linear equations in n variables is a vector space (use the usual addition and scalar multiplication in Rn)."

Intuitively this makes sense to me, but I'm having trouble writing it in a general way that will prove it for any homogeneous system. If someone could help me with that I'd greatly appreciate it.

2. Originally Posted by paulrb
"Show that the solution set of a homogeneous system of linear equations in n variables is a vector space (use the usual addition and scalar multiplication in Rn)."

Intuitively this makes sense to me, but I'm having trouble writing it in a general way that will prove it for any homogeneous system. If someone could help me with that I'd greatly appreciate it.
Let $A$ be an $n\times n$ matrix, the homogenous system is $A\bold{x} = 0$.

If $\bold{x}_1,\bold{x}_2$ are solutions then $A(\bold{x}_1 + \bold{x}_2) = A\bold{x}_1 + A \bold{x}_2 = \bold{0}+\bold{0} = \bold{0}$. Thus, $\bold{x}_1+\bold{x}_2$ lies in the set of zeros.

Now prove that $k\bold{x}$ also lies in the set of solutions.

That will show that the set of solutions is a vector space.