Adjoint of a linear map

Definition

Let $V$ and $W$ be finite-dimensional inner product spaces over $\R$ or $\C$, and let $T \colon V \to W$ be a linear map. The adjoint of $T$ is the unique linear map $T^* \colon W \to V$ such that $$\langle T(v), w \rangle = \langle v, T^*(w) \rangle \qquad \text{for all } v \in V \text{ and } w \in W.$$

Remarks

The existence and uniqueness follow from the Riesz representation theorem. It satisfies $(S + T)^* = S^* + T^*$, $(aT)^* = \bar a T^*$, $(ST)^* = T^* S^*$ and $T^{**} = T$. In orthonormal bases, the matrix of $T^*$ is the conjugate transpose of the matrix of $T$. It satisfies $\ker T^* = (\im T)^\perp$.

Explanations

L1 human

The adjoint lets us move a map from one side of an inner product to the other. For real matrices with the dot product, the adjoint is the transpose: $(Ax) \cdot y = x \cdot (A^T y)$. For complex matrices, it is the conjugate transpose.