Complex conjugate vector space.html

In mathematics, one associates to every complex vector space $V\,$ its complex conjugate vector space $\overline V$ , again a complex vector space. One reason for considering the conjugate vector space is that it allows one to think of antilinear maps as linear maps: an antilinear map from $V\,$ to $W\,$ gives rise to a linear map $\overline V\to W$ , and conversely.

The underlying set of vectors and the addition of $\overline V$ are the same as those of $V\,$ , but the scalar multiplication in $\overline V$ is defined as follows:

to multiply the complex number $\alpha\,$ with the vector $v\,$ in $\overline V$ , take the complex conjugate $\bar \alpha$ of $\alpha\,$ and multiply it with $v\,$ in the original space $V\,$ .

The map $\overline{(\,\cdot\,)}: V\to\overline V$ defined by $\overline v= v$ for all $v\,$ in $V\,$ is then bijective and antilinear. Furthermore, we have $\overline{\overline V}= V$ and $\overline{\overline v}=v$ for all $v\,$ in $V\,$ .

An antilinear map $V \to W$ , for another vector space $W\,$ , is the same thing as a linear map $\overline V\to W$ .

Given a linear map $f:V\to W$ , the conjugate linear map $\overline f:\overline V\to \overline W$ is defined by the formula:

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

The conjugate linear map $\overline f$ is linear. Moreover, the rules $V\mapsto \overline V$ and $f\mapsto\overline f$ define a functor from the category of C-vector spaces to itself.

If $V\,$ and $W\,$ are finite-dimensional and the map $f\,$ is described by the 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 $\overline W$ .

The vector spaces $V\,$ and $\overline V$ have the same dimension over C and are therefore isomorphic as C-vector spaces. However, there is no natural isomorphism from $V\,$ to $\overline V$ .

Complex conjugate vector space.html

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