# Dual (or biorthogonal) bases

While I was reading the paper, Life Beyond Bases: The Advent of Frames (Part I), I noticed a new and important word dual or biorthogonal bases.

**Where does dual base come from?**

**A simple example**

Consider are two independent vectors in . Then is a base. Now consider an arbitrary vector , it can be written as

My question is how to compute . The conventional answer is easy. Let , , then , so the coefficients are . There is no difficulty to get the coefficients. But can we go a little further?

**Go a little further**

We already know and . Hence of course . Write . So we have

**Summary**

Given a base and an arbitrary vector , then

where is the dual base. Note is the projection length of on the basis $\tilde{b_1}$. And . can also be written as

We know . For a orthogonal base, the dual base is itself. Then