Q) Let H be a subset of R^n defined by one single linear equation,

i.e; H={(x1,x2,...,xn) | a1x1+...+anxn+d=0}.Then show that H is a hyperplane of R^n..

[ I was able to prove that "A hyperplane H in R^n is given by a single linear equation,a1x1+...+anxn+d=0"..

(What I did was as H is a hyperplane in R^n, it is an (n-1) dimension affine subspace of R^n. This implies H=W+p,for some p in R^n and W is a linear subspace of R^n of dim(n-1)..Then choose a basis {w1,...,wn-1} of W and extend it a basis {w1,...,wn} of R^n..Then applyin Gram-Schmidth we obtain orthonormal basis {u1,...,un} of R^n..Hence {u1,...,un-1} is an orthonormal basis of W..

This implies W={v in R^n | <v,un>=0}..

Let x belong to H <=>x-p is in W <=> <x-p,un>=0 <=><x,un>-<p,un>=0..

Let un=(a1,...,an) in R^n, Then we get <(x1,...,xn) , (a1,...,an)> - <p,un>=0

This implies a1x1+...+anxn+d=0, where d+ -<p,un>..]

I need help in provin the above posted question on the same lines of the proof i gave in one way..

Thank you.