How do I find a dual basis for all vectors in $R^3$ such that $v_1-3v_2+2v_3=0?$

Question

How do I find a dual basis for all vectors in $R^3$ such that $v_1-3v_2+2v_3=0?$

I know the "regular" basis $B=\{ (3,1,0), (2,0,-1)\}$. But what is the dual basis?

Answer 1

Essentially the same in $R^3$. The components are the same. The dual space is a different geometric entity. Treat what you have as row vectors. The dual is the transpose of those. They will span a space of column vectors. This allows you to define an inner product between entities spanned by both.