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Math Help - Prove the solution set of a system of homogeneous equations is a vector space

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    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.
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    Quote Originally Posted by paulrb View Post
    "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.
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