Posted by Simon on Sunday, November 20, 2011 at 3:09pm.
Find the basis for the following vector space. Please state the dimension of the vector space. S consists of all x in R3 such that x is orthogonal to n=(2,3,2)

Linear Algebra  MathMate, Sunday, November 20, 2011 at 11:47pm
The dimension of vector space R3 is 3.
n=(2,3,2) occupies one of the three dimensions, so the subspace orthogonal to n has a dimension of two, i.e. two vectors span the remaining subspace.
The basis of the remaining subspace can be in many forms. One way is to start with an arbitrary vector, say a1=(1,1,0) and apply GramSchmidt process to transform it into a vector orthogonal to n. After normalization, we get A1=(7,2,10)/(3sqrt(17)).
Note that A1.n=0.
Similarly, if we start with a2=(1,0,1), and apply GramSchmidt process, we get A2=(1,0,1)/sqrt(2).
Note also that A2.n=0, and A1.A2=0.
Thus we have a basis for R3 {n,A1,A2} of which two are orthogonal to n. (n being one of the three vectors of the basis). 
Linear Algebra  MathMate, Monday, November 21, 2011 at 12:28am
... and the basis for the subspace orthogonal to n is {A1,A2}.