Complex conjugate vector space

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In mathematics, the (formal) complex conjugate of a complex vector space $V\,$ is the complex vector space $\overline V$ consisting of all formal complex conjugates of elements of $V\,$ . That is, $\overline V$ is a vector space whose elements are in one-to-one correspondence with the elements of $V\,$ :

$\overline V = \{\overline v \mid v \in V\},$

with the following rules for addition and scalar multiplication:

$\overline v + \overline w = \overline{\,v+w\,}\quad\text{and}\quad\alpha\,\overline v = \overline{\,\overline \alpha \,v\,}.$

Here $v\,$ and $w\,$ are vectors in $V\,$ , $\alpha\,$ is a complex number, and $\overline\alpha$ denotes the complex conjugate of $\alpha\,$ .

In the case where $V\,$ is a linear subspace of $\mathbb{C}^n$ , the formal complex conjugate $\overline V$ is naturally isomorphic to the actual complex conjugate subspace of $V\,$ in $\mathbb{C}^n$ .

[edit] Antilinear maps

If $V\,$ and $W\,$ are complex vector spaces, a function $f\colon V \to W\,$ is antilinear if

$f(v+v') = f(v) + f(v')\quad\text{and}\quad f(\alpha v) = \overline\alpha \, f(v)$

for all $v,v'\in V\,$ and $\alpha\in\mathbb{C}$ .

One reason to consider the vector space $\overline V$ is that it makes antilinear maps into linear maps. Specifically, if $f\colon V \to W\,$ is an antilinear map, then the corresponding map $\overline V \to W$ defined by

$\overline v \mapsto f(v)$

is linear. Conversely, any linear map defined on $\overline V$ gives rise to an antilinear map on $V\,$ .

One way of thinking about this correspondence is that the map $C\colon V \to \overline V$ defined by

$C(v) = \overline v$

is an antilinear bijection. Thus if $f\colon \overline V \to W$ if linear, then then composition $f \circ C\colon V \to W\,$ is antilinear, and vice-versa.

[edit] Conjugate linear maps

Any linear map $f \colon V \to W\,$ induces a conjugate linear map $\overline f \colon \overline V \to \overline W$ , defined by the formula

$\overline f (\overline v) = \overline{\,f(v)\,}.$

The conjugate linear map $\overline f$ is linear. Moreover, the identity map on $V\,$ induces the identity map $\overline V$ , and

$\overline f \circ \overline g = \overline{\,f \circ g\,}$

for any two linear maps $f\,$ and $g\,$ . Therefore, the rules $V\mapsto \overline V$ and $f\mapsto\overline f$ define a functor from the category of complex vector spaces to itself.

If $V\,$ and $W\,$ are finite-dimensional and the map $f\,$ is described by the complex matrix $A\,$ with respect to the bases $\mathcal B$ of $V\,$ and $\mathcal C$ of $W\,$ , then the map $\overline f$ is described by the complex conjugate of $A\,$ with respect to the bases $\overline{\mathcal B}$ of $\overline V$ and $\overline{\mathcal C}$ of $\overline W$ .

[edit] Structure of the conjugate

The vector spaces $V\,$ and $\overline V$ have the same dimension over the complex numbers and are therefore isomorphic as complex vector spaces. However, there is no natural isomorphism from $V\,$ to $\overline V$ . (The map $C\,$ is not an isomorphism, since it is antilinear.)