Duality: Basis vs Coordinate Functions

This applet visualizes a relationship between a basis in a plane and the dual basis. Given a basis

, 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,

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