Dualbasis

In linear algebra, the dual basis is associated with a finite basis of a vector space. Let V be a finite-dimensional vector space over a field F, and let B = {v1, v2, ..., vn} be a basis of V. The dual space V* consists of all linear functionals φ: V → F. The dual basis B* = {f1, f2, ..., fn} is the corresponding basis of V* defined by the property f_i(v_j) = δ_ij, where δ_ij is the Kronecker delta.

Construction and interpretation: For any φ in V*, there is a unique expression φ = ∑_{i=1}^n φ(v_i) f_i. Thus the functional φ is determined by its values on the basis vectors, and these values are the coordinates of φ relative to the dual basis. Conversely, for any vector x in V with coordinates x = ∑ a_i v_i relative to B, applying the dual basis gives f_i(x) = a_i. Therefore the dual basis extracts the coordinate components of vectors with respect to the original basis.

Change of basis and matrices: If the basis B is represented by a matrix with columns v1, ..., vn, then the dual basis corresponds to the rows of the inverse transpose when switching to coordinates in the dual space. In particular, the map from V to F^n that sends x to (f_1(x), ..., f_n(x)) is the coordinate map relative to B*.

Example: In R^2 with the standard basis e1 = (1,0), e2 = (0,1), the dual basis is e^1, e^2, where e^1(x1, x2) = x1 and e^2(x1, x2) = x2. If a different basis, such as v1 = (1,1) and v2 = (1,-1), is chosen, the corresponding dual functionals are f1(x,y) = (x+y)/2 and f2(x,y) = (x−y)/2.

Applications: The dual basis is fundamental for coordinate extraction, linear functional representation, and formulating bilinear pairings. It also clarifies the relationship between a basis and its induced functionals in the dual space.