I have two vectors $u$ and $v$ in $\Bbb{R}^4$ that span a vector space. I need to find the basis of its' orthogonal complement where the inner product is not the dot product. How would I do this?
I know that if it were the dot product, I could use the result of $Row(A)^\bot = Nul(A)$ and use row reduction to find the null space and hence the basis of the orthogonal complement. However, I do not know how to do this when the inner product is not the dot product. Does it require use of orthonormalisation? Any help would be much appreciated
Edit: As the inner product was of the form: $$\lt u,v \gt = a_1\cdot u_1 \cdot v_1 + a_2\cdot u_2 \cdot v_2 + a_3\cdot u_3 \cdot v_3 + a_4\cdot u_4 \cdot v_4; a_1, a_2, a_3, a_4 \in \Bbb{R} $$ I multiplied the columns of a matrix $A$ containing $u$ and $v$ and two empty rows by $a_1, a_2, a_3$ and $a_4$ respectively. Then I used back substitution to find the null space and the basis for the null space, and used it to find the orthogonal complement. Does it work, and if it does, can it be used for other inner product spaces or is it just a coincidence of this inner product space that it works?