Duality: Basis vs Coordinate Functions

This applet visualizes a relationship between a basis in a plane and the dual basis. Given a basis  in , the dual basis consists of two coordinate functionals , defined as  and . The red coordinate grid consists of level curves  .  The nullspace  Nul  is just the line . Denote by "" the dot product with respect to the standard basis  (not shown). Suppose a linear functional   is written in the standard basis as . Its  gradient vector    has the property .  Note that   is orthogonal to  Nul . Thus, we can visualize the dual basis  as a pair of vectors   and .
Tasks 1. Drag the endpoints of the black vectors and observe how the dual basis changes. 2. Position  in such way that  would have length 1. HINT: In that case, the projection of  onto  will be  itself.  3. Suppose   and . What are the coordinates of the blue vectors then?  HINTS: Write the standard basis vectors as linear combinations of  first. Alternatively, the blue vectors  must satisfy to the  4 equations . In matrix form, .