Dual basis ($e_i^{\ast }(e_j)={\delta }_{ij}$)

For a basis $e_1,\dots ,e_n$ of a finite-dimensional vector space $V$, its dual basis consists of linear functionals $e_1^{\ast },\dots ,e_n^{\ast }$ satisfyingThe Kronecker delta means that each rule gives one on its own basis vector and zero on the others. These rules extract coordinates:Applying $e_i^{\ast }$ to a linear combination removes every term except the $i$th. Consequently the dual basis is a basis of the dual space, and every linear functional has expansion $\phi ={\sum }_i\phi (e_i)e_i^{\ast }$. ^[ref].

Coordinate extraction works for a nonorthogonal basis too. With $b_1=(1,0)$ and $b_2=(1,1)$, the dual basis isThe checks $b_1^{\ast }(b_1)=1$, $b_1^{\ast }(b_2)=0$, $b_2^{\ast }(b_1)=0$, and $b_2^{\ast }(b_2)=1$ fix the two rules. For $v=(3,1)$ they give coordinates $(2,1)$, so $v=2b_1+b_2$. The usual dot products with $b_1,b_2$ instead give $3,4$ and do not extract these coordinates: simple projection formulas depend on an orthonormal basis.

The same idea explains polynomial interpolation. On polynomials of polynomial degree at most $n$, evaluation at distinct interpolation nodes $x_0,\dots ,x_n$ gives the dual basis to the Lagrange cardinal polynomials, because ${\ell }_i(x_j)={\delta }_{ij}$. Samples are coordinates in this basis, so the reconstruction $p={\sum }_{i}p(x_i){\ell }_i$ is a dual-basis expansion. Differentiation weights from nodal evaluations and factorial monomial basis for polynomial differentiation apply the same principle to different measurements.

For $p(X)=a+bX+c{X}^2$, the three point evaluation functionals at $0,1,-1$ form a basis of the dual space of real polynomials of degree at most two. The displayed recovery formulas express the coordinate functionals, hence the dual basis to $1,X,X^2$, as linear combinations of these evaluations. More generally distinct-node polynomial interpolation identifies values with coefficient data through an invertible Vandermonde matrix.

For the basis $v_j=e_1+\cdots +e_j$ obtained from a basis $e_1,\dots ,e_n$, the dual basis is obtained by successive differences of the original dual basis. Indeed, if $x={\sum }_j{c}_j{v}_j$, its original coordinates are $x_i={\sum }_{j\ge i}{c}_j$. Hence $c_i=x_i-x_{i+1}$ for $i<n$, and $c_n=x_n$. These are precisely the new coordinate linear functionals. This gives an elementary example of the inverse relation between change of basis and covector transformation under a change of basis.

A coordinate functional extracts one coefficient in a specified finite basis. It is an element of that basis's dual basis, rather than a coordinate rule independent of the basis. For the standard basis of $F^n$, the $i$th coordinate functional sends $(a_1,\dots ,a_n)$ to $a_i$.

On polynomials of degree at most $n$, the derivative-evaluation functionals $e_j(\varphi )={\varphi }^{(j)}(0)$ are dual to $p_j=x^j/j!$, since $e_i(p_j)={\delta }_{ij}$. Differentiation maps $p_j$ to $p_{j-1}$, with $p_0$ mapped to zero. Its matrix is a single nilpotent Jordan block with ones above the diagonal, and its dual has the transpose matrix.