How are anti-unitary operators applied?

I was reading about anti-unitary operators from Wikipedia. They give an example of an anti-unitary operator:

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.

The antiunitary operator is an operator on the Hilbert space. Thus, it is nonsense to say that "the operator $K$ apply on complex number". Nevertheless, it can be shown that $U\alpha |\phi\rangle=\alpha^*U |\phi\rangle$, where $U$ is an antiunitary operator. $U|\phi\rangle$ should not be confused with $\langle\phi|$, the complex conjugate of $|\phi\rangle$.