Linear Algebra

Dual Basis and Linear Functionals

Visualize dual basis, linear functionals, and dual expansion coefficients.

By Riano · Published · Updated · Report a correction

Dual Basis and Linear Functionals

Concept Overview

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 fi(bj) = δij, where δij 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.

Worked Example

Let's calculate the dual basis and evaluate a linear functional for a specific basis and vector in ℝ2. Suppose our basis matrix B has columns b1 = [1, 1]T and b2 = [1, -1]T. We have a vector x = [3, 1]T and a linear functional g = [2, -1]. Here is the step by step process to evaluate the coordinates and the dual expansion coefficients.

  1. Find the Dual Basis (Rows of B-1):
    B = [[1, 1], [1, -1]].
    The determinant is det(B) = (1)(-1) - (1)(1) = -2.
    B-1 = (-1/2) * [[-1, -1], [-1, 1]] = [[1/2, 1/2], [1/2, -1/2]].
    The dual basis consists of the rows of B-1:f1 = [1/2, 1/2] and f2 = [1/2, -1/2].
  2. Verify Dual Basis Properties:
    f1(b1) = (1/2)(1) + (1/2)(1) = 1.
    f1(b2) = (1/2)(1) + (1/2)(-1) = 0.
    f2(b1) = (1/2)(1) + (-1/2)(1) = 0.
    f2(b2) = (1/2)(1) + (-1/2)(-1) = 1.
  3. Coordinates of x in Basis B:
    The coordinates are given by ci = fi(x).
    c1 = f1(x) = [1/2, 1/2] · [3, 1] = 3/2 + 1/2 = 2.
    c2 = f2(x) = [1/2, -1/2] · [3, 1] = 3/2 - 1/2 = 1.
    So, [x]B = [2, 1]T. Check: 2*b1 + 1*b2 = [2, 2] + [1, -1] = [3, 1] = x.
  4. Expansion of g in Dual Basis B*:
    The coefficients are given by di = g(bi).
    d1 = g(b1) = [2, -1] · [1, 1] = 2(1) + (-1)(1) = 1.
    d2 = g(b2) = [2, -1] · [1, -1] = 2(1) + (-1)(-1) = 3.
    So, g = 1*f1 + 3*f2 = 1*[1/2, 1/2] + 3*[1/2, -1/2] = [1/2, 1/2] + [3/2, -3/2] = [2, -1].
  5. Evaluate g(x):
    Directly: g(x) = [2, -1] · [3, 1] = 2(3) + (-1)(1) = 6 - 1 = 5.
    Using expansion: g(x) = d1c1 + d2c2 = (1)(2) + (3)(1) = 2 + 3 = 5.

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.

Limitations

This visualization focuses on 2D vector spaces (ℝ2) to clearly illustrate the geometric intuition of level sets for linear functionals. In higher dimensions, linear functionals represent hyperplanes (e.g., planes in 3D), which are harder to visualize alongside vectors. Every input must be an integer from -3 to 3. Empty, nonfinite, fractional or out-of-range entries show an error and remove dependent results. With these integer bounds the determinant is computed exactly, so singularity is tested at zero. Dual rows and scalar results are displayed to six decimal places, while reconstruction uses unrounded arithmetic. The visualization of covectors as level lines is a simplified graphical representation; in formal differential geometry, covectors are 1-forms, but drawing level sets (where the functional evaluates to integer values) effectively captures their linear nature on a vector space.

References

For further information and resources:

Experience it interactively

Adjust parameters, observe in real time, and build deep intuition with Riano’s interactive Dual Basis and Linear Functionals module.

Try Dual Basis and Linear Functionals on Riano →

More in Linear Algebra

Bezier Curves

Parametric curves for computer graphics.

Matrix Multiplication

Visualize how 2x2 matrices transform space and combine through multiplication.

Dot Product & Projection

Visualizing vector projection and orthogonality.