Dual Basis and Linear Functionals

In linear algebra, a linear functional is a linear map from a vector space to its field of scalars (e.g., from ℝ² to ℝ). The set of all linear functionals on a vector space V forms a new vector space called the dual space, denoted V*. Given a basis B = {b₁, b₂} for V, there is a natural corresponding basis for the dual space, called the dual basis B* = {f₁, f₂}, defined by the relation f_i(b_j) = δ_ij, where δ_ij is the Kronecker delta (1 if i=j, 0 otherwise). In matrix terms, if the vectors of B are the columns of a matrix, the linear functionals of the dual basis B* are the rows of its inverse matrix.

Experiment

Explore the dual basis using the interactive visualization:

1. Open the Dual Basis and Linear Functionals module.
2. The default values match our worked example: B11=1, B12=1, B21=1, B22=-1. x1=3, x2=1, g1=2, g2=-1.
3. Observe the canvas: The primal basis vectors b₁ and b₂ are shown as vectors (arrows). The linear functionals f₁, f₂, and g are shown as sets of level lines. Notice that the level lines of f₁ are parallel to b₂, and the level lines of f₂ are parallel to b₁.
4. Change B12 to 2 and B21 to 0 (making B an upper triangular matrix). Notice how the dual basis rows (functionals) change their orientation, and how the level lines adjust.
5. Try setting B11=1, B12=2, B21=1, B22=2. This makes the basis vectors linearly dependent (determinant is 0). The module will display an explicit "Dependent basis: no unique dual basis" message, as the inverse matrix B^-1 does not exist.