I was reading about [anti-uintary operators][1] from wikipedia. They give an example of an anti-unitary operator : 




  ![enter image description here][2]
<br><br>
were $K$ is complex conjugate operation. $\sigma_y$ is defined with respect to two orthonormal vectors spanning a 2-D Hilbert space $H$ say $|0\rangle$ and $|1\rangle$. Now if I have an initial vector as $\alpha|0\rangle+\beta|1\rangle$ would the operator $K$ apply only on complex number $\alpha,\beta$ ( result being $\alpha^*|0\rangle+\beta^*|1\rangle$  )  or on the vector $|0\rangle$ and $|1\rangle$
aslo ? If it does what does it mean to take complex conjugate of $|0\rangle$ and $|1\rangle$ ? My doubt arises because anti-unitaries not being linear can't be expressed only in terms of matrices.

  [1]: http://en.wikipedia.org/wiki/Antiunitary_operator
  [2]: https://i.sstatic.net/sIWpB.png