The dual space records all linear measurements of vectors. Adjoints move linear maps from vectors to measurements, or from one side of an inner product to the other.
Linear functionals and dual bases
Definition. The dual space of \(V\) is \[ V^*=\{\varphi:V\to F:\varphi\text{ linear}\}. \] If \(B=(v_1, \ldots,v_n)\) is a basis of \(V\), the dual basis \((v^1, \ldots,v^n)\) is defined by \(v^i(v_j)=\delta_{ij}\).
Proposition. If \(V\) is finite-dimensional, then \(\dim V^*=\dim V\), and every \(\varphi\in V^*\) has the unique expansion \[ \varphi=\sum_{i=1}^n\varphi(v_i)v^i. \]
Annihilators
Definition. If \(U\subset V\), its annihilator is \[ U^\circ=\{\varphi\in V^*: \varphi(u)=0\text{ for all }u\in U\}. \] It is a subspace of \(V^*\).
Proposition. If \(V\) is finite-dimensional and \(U\subset V\) is a subspace, then \[ \dim U^\circ=\dim V- \dim U. \] Moreover, \((V/U)^*\cong U^\circ\).
Proof. Extend a basis \((u_1, \ldots,u_k)\) of \(U\) to a basis \((u_1, \ldots,u_k,v_{k+1}, \ldots,v_n)\) of \(V\). A functional vanishes on \(U\) exactly when its first \(k\) dual coordinates are zero, giving dimension \(n-k\). A functional on \(V/U\) pulls back along \(V\to V/U\) to a functional on \(V\) that vanishes on \(U\), and this gives the isomorphism.
Transpose maps
Definition. For a linear map \(T:V\to W\), the dual or transpose map is \[ T':W^* \to V^*, \qquad T'(\psi)=\psi\circ T. \] It reverses arrows.
Proposition. In dual bases, the matrix of \(T':W^*\to V^*\) is the transpose of the matrix of \(T:V\to W\). Also, \[ \ker T'=(\operatorname{im}T)^\circ, \qquad \operatorname{im}T'=(\ker T)^\circ \] in finite dimension.
Adjoints under an inner product
Definition. If \(V,W\) are inner product spaces, the Hilbert adjoint \(T^*:W\to V\) is the unique map satisfying \[ \langle Tv,w\rangle_W=\langle v,T^*w\rangle_V. \] In orthonormal bases, its matrix is the conjugate transpose.
Double dual
Theorem. The evaluation map \[ \eta:V\to V^{**}, \qquad \eta(v)(\varphi)=\varphi(v), \] is always injective. If \(V\) is finite-dimensional, it is an isomorphism.
Proof. If \(v\ne0\), extend \(v\) to a basis and take the dual coordinate functional that sends \(v\) to \(1\); hence \(\eta(v)\ne0\). In finite dimension, \(\dim V=\dim V^*=\dim V^{**}\), so an injective map \(V\to V^{**}\) is bijective.
Hyperplanes and linear equations
Proposition. A proper subspace \(H\subset V\) has codimension one if and only if \(H=\ker\varphi\) for some nonzero functional \(\varphi\in V^*\). Thus nonzero linear functionals are exactly homogeneous linear equations whose solution sets are hyperplanes.
Remark. The algebraic dual \(V^*\) and the inner-product identification \(V\cong V^*\) should not be confused. An inner product turns covectors into vectors, but the construction depends on the chosen inner product.
The coordinate dual basis
Example. If \(B=(v_1,\ldots,v_n)\) is a basis of \(V\), the dual basis \(B^*=(\varepsilon^1,\ldots,\varepsilon^n)\) is defined by \(\varepsilon^i(v_j)=\delta_{ij}\). For \(v=\sum_j a_jv_j\), one has \(\varepsilon^i(v)=a_i\). Thus dual vectors are coordinate-reading functions once a basis is chosen.
Calculation. If \(T:V\to W\) has matrix \(A\) in bases \(B,C\), then \(T':W^*\to V^*\) has matrix \(A^\top\) in the dual bases. Indeed, the coefficient of \(\varepsilon^i\) in \(T'\eta^j\) is \(\eta^j(Tv_i)\), the \((j,i)\)-entry of \(A\).