An _anti-linear [map](Maps.md)_ or _anti-linear transformation_ is a map, $f:\mathcal{V}\rightarrow \mathcal{W}$, between [complex vector spaces](Complex%20vector%20spaces.md), $\mathcal{V}$ and $\mathcal{W}$ such that, 1) $f(a\mathbf{u}+b\mathbf{v}) = a^*f(\mathbf{u})+ b^*f(\mathbf{v})$ 2) $f(c\mathbf{u}) = c^* f(\mathbf{u})$ where $\mathbf{u},\mathbf{v} \in \mathcal{V}$ and $a.b,c \in \mathbb{C}.$ #MathematicalFoundations/Algebra/AbstractAlgebra/LinearAlgebra