Duality: Basis vs Coordinate Functions
- Eugene Yablonski
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, .